arxiv: 2605.11370 · v1 · submitted 2026-05-12 · 🌌 astro-ph.HE

Recognition: 2 theorem links

· Lean Theorem

Transonic accretion and the analogue gravity in multi-component elliptical galaxies hosting pseudo-Schwarzschild black holes

Ripon Sk, Sangita Chatterjee, Sankhasubhra Nag

Pith reviewed 2026-05-13 02:20 UTC · model grok-4.3

classification 🌌 astro-ph.HE

keywords transonic accretionmulti-component galactic potentialpseudo-Schwarzschild black holesvertical equilibrium discshock formationanalogue gravitycritical pointselliptical galaxies

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The pith

The galactic potential in elliptical galaxies strongly affects critical points, shock locations, and acoustic surface gravity in transonic accretion flows for all tested pseudo-Schwarzschild black hole potentials.

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

This paper investigates low-angular-momentum accretion flows onto non-rotating black holes embedded in multi-component elliptical galaxies. It incorporates stellar, dark matter, and hot gas contributions to the gravitational potential alongside the black hole's pseudo-Schwarzschild potential. Using the vertical equilibrium disc model and two thermodynamic equations of state, the analysis shows that the surrounding galactic mass distribution changes the locations where the flow becomes supersonic, the range of parameters allowing shocks, and the resulting flow properties after shocks. These changes also modify the acoustic surface gravity, which is relevant for analogue gravity studies. A sympathetic reader would care because real accretion in galaxies occurs within extended mass distributions rather than in isolation, potentially altering predictions for observable emissions and flow stability.

Core claim

Through the analysis of transonic behaviour and eigenvalue-based critical point classification in the vertical equilibrium geometry, the work demonstrates that for all selected black hole potentials, the multi-component galactic potential profoundly influences the locations of critical points, the shock-allowed parameter space, shock-location, shock-driven flow variables, and acoustic surface gravity.

What carries the argument

The vertical equilibrium disc model with a static multi-component galactic potential (stellar, dark matter, hot gas) and various pseudo-Schwarzschild black hole potentials.

Load-bearing premise

The vertical equilibrium disc model combined with pseudo-Schwarzschild potentials and a static multi-component galactic potential sufficiently captures the essential dynamics of real, low-angular-momentum accretion flows.

What would settle it

Comparing the predicted shock locations and critical point positions from this model against results from three-dimensional general relativistic hydrodynamic simulations of accretion in a multi-component galactic potential.

Figures

Figures reproduced from arXiv: 2605.11370 by Ripon Sk, Sangita Chatterjee, Sankhasubhra Nag.

**Figure 1.** Figure 1: In this figure we show the variation of the specific energy of the flow with the location of critical points for different values of specific angular momentum, using ΦBH 𝑖 for the black hole pseudo potential and ΦGal 𝑖 for the galactic potential. The curves are drawn for different values of 𝜆: (a) 𝜆 = 1.60, 1.66, 1.70, 1.76; (b) 𝜆 = 1.40, 1.50, 1.60, 1.70; (c) 𝜆 = 0.95, 1.00, 1.05, 1.10; and (d) 𝜆 = 1.20, … view at source ↗

**Figure 2.** Figure 2: We plot different regions in the parameter space of specific energy (E) and specific angular momentum (𝜆), corresponding to the number and nature of critical points in adiabatic accretion flows. The diagram shows the classification of these regions for the VE disc model. The parameter space is plotted for ΦBH 𝑖 (dashed line) considering BH pseudo Schwarzschild potential, and for ΦGal 𝑖 representing the gal… view at source ↗

**Figure 3.** Figure 3: We present here the phase portrait of adiabatic transonic accretion where the horizontal axis represent the 𝑙𝑜𝑔10𝑟 and the vertical axis represent the mach number 𝑀 = 𝑣 𝑐𝑠 . 𝑟 is the radial distance from the central black hole, 𝑣 and 𝑐𝑠 are the flow veocity and sound speed respectively. The left panels show the plots using BH pseudo potential ΦBH 𝑖 , while right panels include galactic potential ΦGal 𝑖 wit… view at source ↗

**Figure 4.** Figure 4: In this figure, we present the nature of critical points under the BH pseudo potentials ΦBH 𝑖 in left panel and galactic potential ΦGal 𝑖 in right panel with galactic parameter Υ𝐵 = 100. A saddle point is indicated by Ω2 > 0 in all subplots, while a center-type point is indicated by Ω2 < 0. A typical sequence is depicted in each subplot: a single saddle point at low 𝜆; the emergence of a centre and an addi… view at source ↗

**Figure 5.** Figure 5: In this figure we compare the shock-permitting regions in the parameter space (E–𝜆) with the galactic potential ΦGal 𝑖 embedded in four pseudo Schwarzschild potentials. We assume two values of the galactic parameter given as, solid lines correspond to Υ𝐵 = 14, while dashed lines correspond to Υ𝐵 = 100. The figure highlights the influence of Υ𝐵 on the size and location of the shock-permitting regions, illus… view at source ↗

**Figure 6.** Figure 6: In this figure, we compare the shock-permitting regions in the temperature–angular momentum (𝑇–𝜆) parameter space with the galactic potential ΦGal 𝑖 with the chosen pseudo Schwarzschild potential. We assume two values of the galactic parameter given as, solid lines correspond to Υ𝐵 = 14, while dashed lines correspond to Υ𝐵 = 100. The figure highlights the comparative behavior and the influence of Υ𝐵 on sho… view at source ↗

**Figure 7.** Figure 7: In this figure we present the variation of shock strength for the galactic parameter: Υ𝐵 = 14, Υ𝐵 = 33, Υ𝐵 = 100, and Υ𝐵 = 390 considering four pseudo Schwarzschild potential. The results are obtained for E, 𝜆] = [0.001, 1.80] in panel (a), for E, 𝜆] = [0.001, 1.17] in panel (b), for E, 𝜆] = [0.001, 1.17] in panel (c) and for E, 𝜆] = [0.001, 1.42] in panel (d) respectively. Each panel shows the variation o… view at source ↗

**Figure 8.** Figure 8: This figure present the variation of shock location for the galactic potential ΦGal 𝑖 considering the four pseudo-Schwarzschild potential as described. We consider Υ𝐵 = 14, Υ𝐵 = 33, Υ𝐵 = 100, and Υ𝐵 = 390 to plot this figure. The figure illustrates how the position of shock formation in axisymmetric transonic flows is influenced by the strength of the galactic potential and the associated pseudo Schwarzsch… view at source ↗

**Figure 9.** Figure 9: We plot this figure to show the variation of the acoustic surface gravity (𝜅) for the galactic potential ΦGal 𝑖 with four pseudo Schwarzschild potential. The analysis is carried out for a range of galactic parameters: Υ𝐵 = 14, 33, 100, and 390. The figure illustrates how 𝜅 responds to changes in the galactic environment, reflecting the combined influence of black hole gravity and the extended galactic pote… view at source ↗

read the original abstract

Low-angular-momentum, axisymmetric, inviscid accretion flows onto a black hole have been studied using the vertical equilibrium disc model, considering multiple pseudo-Schwarzschild potentials and two thermodynamic equations of state. A multi-component galactic potential-representing stellar, dark matter, and hot-gas contributions-is incorporated to assess environmental effects on the accretion dynamics. In our earlier work, it is found that the effect of multi-component galactic potential on the accretion flow onto a rotating black hole under similar framework of analysis, significantly varies over different standard disc models, being most pronounced in the vertical equilibrium (VE) disc model. Thus it may be interesting to find whether such variation occur for different choices of pseudo potentials too. To begin with, in this work we consider accretion flow onto a non-rotating blackhole with VE geometry. Through the analysis of transonic behaviour and eigenvalue-based critical point classification, we demonstrate that, for all selected black hole potentials, the galactic potential profoundly influences the locations of critical points, the shock-allowed parameter space, shock-location, shock-driven flow variables, and acoustic surface gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the authors' rotating-BH galactic-potential study to the non-rotating case and shows the same qualitative shifts in critical points and shocks, but stays inside the existing VE pseudo-potential framework.

read the letter

The main point is that a static multi-component galactic potential moves the sonic points, changes the range of parameters that allow shocks, and alters the acoustic surface gravity for non-rotating pseudo-Schwarzschild flows in vertical equilibrium. The effect appears across the potentials they test and for both equations of state they consider, matching the pattern they found earlier for rotating holes. That consistency is the useful new piece here. They classify critical points with the usual eigenvalue method and locate shocks from the conserved quantities, so the numerics follow the standard procedure without obvious gaps in the logic. The work is careful in checking that the galactic term is not an artifact of one particular potential choice. The limitation is that everything remains inside the same set of approximations: pseudo-potentials rather than full GR, vertical equilibrium rather than a full disc structure, and a fixed galactic potential with no back-reaction or time dependence. The size of the shifts depends on how the stellar, dark-matter, and gas components are normalized, and those normalizations are not varied widely. No comparison to observations or to time-dependent simulations is offered, so the claim of “profound” influence is model-internal. This is a paper for people already working on low-angular-momentum transonic flows and analogue gravity in galaxies. A reader who follows the vertical-equilibrium literature or the authors’ earlier rotating case will find the specific numbers and the cross-check across potentials worth having. It is coherent and reproducible enough on its own terms to go to a serious referee rather than being rejected outright.

Referee Report

1 major / 2 minor

Summary. The manuscript studies low-angular-momentum, axisymmetric, inviscid accretion onto non-rotating black holes in the vertical-equilibrium disc model. It employs several pseudo-Schwarzschild potentials and two equations of state, augments the effective potential with a static multi-component galactic term (stellar + dark matter + hot gas), and analyzes transonic critical points via eigenvalue classification, shock existence, post-shock variables, and acoustic surface gravity. The central claim is that the galactic potential profoundly modifies all of these quantities for every black-hole potential examined.

Significance. If the numerical results are robust, the work supplies a controlled demonstration that galactic-scale potentials cannot be neglected in models of low-angular-momentum accretion in elliptical galaxies. By repeating the analysis across multiple pseudo-potentials and both polytropic and isothermal closures, the authors provide an internal consistency check that the reported shifts are not artifacts of a single choice. The extension from the authors’ earlier rotating-BH study to the non-rotating VE case is a natural and useful increment.

major comments (1)

[Abstract] Abstract and §3 (or equivalent results section): the repeated assertion that the galactic potential “profoundly influences” critical-point locations, shock parameter space, shock location, and acoustic surface gravity for “all selected black hole potentials” is the load-bearing claim, yet the abstract supplies no numerical measure of the shift (e.g., fractional change in r_c or r_sh) relative to the no-galactic-potential baseline. Without such quantification or a table comparing the two cases, the strength of the influence remains unverified.

minor comments (2)

The title advertises “analogue gravity” but the abstract only mentions acoustic surface gravity in passing; a brief sentence linking the computed surface gravity to analogue-horizon properties would clarify the connection.
Notation for the multi-component galactic potential (stellar, DM, hot-gas terms) should be defined once in the text and used consistently; the abstract introduces the components but does not label them.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the work and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and §3 (or equivalent results section): the repeated assertion that the galactic potential “profoundly influences” critical-point locations, shock parameter space, shock location, and acoustic surface gravity for “all selected black hole potentials” is the load-bearing claim, yet the abstract supplies no numerical measure of the shift (e.g., fractional change in r_c or r_sh) relative to the no-galactic-potential baseline. Without such quantification or a table comparing the two cases, the strength of the influence remains unverified.

Authors: We agree that explicit numerical quantification strengthens the central claim. The manuscript already contains figures (e.g., Figs. 1–4) that illustrate the shifts in critical-point locations, shock positions, and acoustic surface gravity when the multi-component galactic potential is included versus the baseline case for each pseudo-Schwarzschild potential and both equations of state. However, we acknowledge that a direct side-by-side numerical comparison is not tabulated and that the abstract lacks specific fractional changes. In the revised manuscript we will (i) insert a compact comparison table in §3 listing r_c, r_sh, post-shock variables, and acoustic surface gravity with and without the galactic term for representative parameter values, and (ii) add one or two quantitative statements (e.g., “shifts of 15–40 % in r_c and up to 60 % in r_sh”) to the abstract. These additions will make the magnitude of the influence immediately verifiable while preserving the existing visual and analytical results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from independent numerical solutions

full rationale

The paper's derivation proceeds via standard transonic flow equations in the vertical-equilibrium framework, constructing an effective potential that adds the multi-component galactic term to established pseudo-Schwarzschild potentials, then solving for critical-point locations, eigenvalue classification, shock parameter space, and acoustic surface gravity through direct numerical integration. The sole self-reference to prior rotating-BH work is confined to the introduction as motivation for selecting the VE geometry; it supplies no fitted parameters, no uniqueness theorem, and no load-bearing input that the current non-rotating calculations reduce to. All reported influences are therefore new outputs of the augmented equations rather than algebraic or statistical re-statements of earlier results.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard domain assumptions of accretion theory plus the specific modeling choices stated in the abstract. No new entities are postulated.

axioms (3)

domain assumption Vertical equilibrium holds for the disc geometry throughout the flow.
Invoked to close the set of equations for the axisymmetric inviscid flow.
domain assumption Pseudo-Schwarzschild potentials provide an adequate approximation to the spacetime geometry near a non-rotating black hole.
Multiple such potentials are adopted to test robustness.
domain assumption The multi-component galactic potential can be treated as static and spherically symmetric.
Used to assess environmental effects on accretion dynamics.

pith-pipeline@v0.9.0 · 5500 in / 1497 out tokens · 56410 ms · 2026-05-13T02:20:23.137618+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We adopt the vertical equilibrium disc model... Φ_Gal_i(r) = Φ_BH_i(r) + Φ_star(r) + Φ_gas(r) + Φ_DM(r)... critical point conditions... Ω² = QR − PS... acoustic surface gravity κ
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
eigenvalue-based critical point classification... saddle-center bifurcations

Reference graph

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