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arxiv: 2604.20344 · v1 · submitted 2026-04-22 · 🌌 astro-ph.GA · astro-ph.CO

The core of the problem: Physical limits of the core-S\'ersic model

Maarten Baes (UGent) This is my paper

Pith reviewed 2026-05-10 00:06 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.CO
keywords core-Sérsic modelstellar density profilesgalactic coresearly-type galaxiesnumerical deprojectionmonotonicitymass deficits
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The pith

Core-Sérsic models with sharp transitions generate non-monotonic densities that cannot describe real galaxies

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

The core-Sérsic model combines an inner power-law core with an outer Sérsic profile to fit the centers of massive elliptical galaxies. Numerical deprojections of this model reveal that sufficiently sharp transitions between the two components produce intrinsic density profiles that increase with radius in places. The paper maps out, for each choice of core slope, Sérsic index, and break-radius ratio, the largest transition sharpness still compatible with a strictly decreasing three-dimensional density. This excludes a substantial fraction of the parameter space that observers and simulators have treated as valid, including the infinite-sharpness limit often used in practice.

Core claim

Numerical deprojections demonstrate that large values of the transition parameter α in the core-Sérsic model always generate non-monotonic intrinsic density profiles. For any fixed set of structural parameters γ, m, and R_e/R_b there exists a critical value α_crit above which monotonicity is violated, so that a non-negligible portion of the formally allowed parameter space, including the sharp-transition limit α → ∞, is physically ruled out.

What carries the argument

The critical transition parameter α_crit, identified through numerical deprojections, that marks the onset of non-monotonic intrinsic density for each combination of core slope γ, Sérsic index m, and size ratio R_e/R_b.

If this is right

  • A fraction of the parameter combinations routinely adopted for the core-Sérsic model cannot represent physical stellar systems.
  • Measurements of core sizes and mass deficits in massive ellipticals must be restricted to the admissible range of α.
  • Dynamical models built on core-Sérsic density profiles should employ only parameter sets that preserve monotonicity.
  • Comparisons of observed core properties with simulations of supermassive black hole binary evolution are limited to the physically allowed region of parameter space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Observers fitting core-Sérsic models may need to impose α ≤ α_crit as an explicit prior or post-fit filter.
  • The dependence of α_crit on structural parameters could be used to predict which galaxy types are more likely to show admissible sharp cores.
  • Alternative functional forms with built-in monotonicity guarantees might be developed to replace the core-Sérsic model in regimes where α_crit is small.

Load-bearing premise

A realistic stellar system must have a strictly monotonic, decreasing intrinsic density profile and numerical deprojection recovers that profile without significant projection or seeing effects.

What would settle it

A galaxy whose best-fit core-Sérsic parameters have α larger than the α_crit computed for its measured γ and m, yet whose independently deprojected density profile remains strictly monotonic, would contradict the central claim.

Figures

Figures reproduced from arXiv: 2604.20344 by Maarten Baes (UGent).

Figure 1
Figure 1. Figure 1: Surface-brightness profiles (left), intrinsic luminosity density profiles (middle), and logarithmic density slopes (right) for sharp-transition core–Sérsic models (α → ∞). All models have a total luminosity of L = 1, a break radius of Rb = 1, and an effective radius of Re = 5. In the top row the Sérsic index is fixed to m = 4, while the inner power-law slope, γ, is varied as indicated; in the bottom row th… view at source ↗
Figure 2
Figure 2. Figure 2: Intrinsic luminosity density profiles (left) and logarithmic density slopes γ(r) (right) for a sequence of core–Sérsic models with increasingly smooth transitions. All models share the same structural parameters as in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

The core-S\'ersic model is the standard tool for describing partially depleted stellar cores in massive early-type galaxies, yet its physical admissibility has rarely been examined. Using numerical deprojections, we show that many formally allowed parameter combinations cannot represent realistic stellar systems: sharp transitions between the inner power-law core and the outer S\'ersic profile (large $\alpha$) always generate non-monotonic intrinsic density profiles. We identify, for each set of structural parameters $(\gamma, m, R_{\text{e}}/R_{\text{b}})$, a critical transition parameter, $\alpha_{\text{crit}}$, above which monotonicity is violated. This threshold systematically depends on the core slope and S\'ersic index, implying that a fraction of the commonly used parameter space, including the widely adopted sharp-transition limit $\alpha\rightarrow\infty$, is physically ruled out. These constraints have important consequences for measuring core sizes and mass deficits in massive ellipticals, for constructing dynamical models, and for comparing observations with simulations of supermassive black hole binary evolution.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims that the core-Sérsic model produces non-monotonic intrinsic density profiles for large values of the transition sharpness parameter α, for any combination of core slope γ, Sérsic index m, and R_e/R_b ratio. Using numerical Abel deprojections, it identifies a critical α_crit for each parameter set above which monotonicity fails, thereby ruling out the commonly adopted sharp-transition limit α → ∞ and a portion of the otherwise allowed parameter space as unphysical for realistic stellar systems.

Significance. If the numerical results are robust, the work imposes concrete physical limits on a standard model used throughout studies of massive early-type galaxies. This would directly affect core-size and mass-deficit measurements, the construction of dynamical models, and comparisons between observations and simulations of supermassive black hole binary scouring. The systematic dependence of α_crit on γ and m supplies a practical fitting constraint. The paper is credited for testing the model's physical admissibility through direct numerical evaluation rather than analytic approximation alone.

major comments (2)
  1. [Numerical deprojection procedure] The central result—that α > α_crit always yields non-monotonic ρ(r)—rests entirely on numerical evaluation of the Abel inversion of the projected core-Sérsic profile. The manuscript provides no description of the quadrature scheme, radial grid, handling of the integrable singularity, or convergence tests as α increases and dI/dR develops a near-discontinuity. This is load-bearing because the inversion integral is known to be sensitive to steep gradients; without error budgets or cross-checks against independent methods or analytic limits, the reported non-monotonicity and α_crit values could be numerical artifacts.
  2. [Results on α_crit] The abstract states that monotonicity violation occurs 'always' for large α, yet no explicit demonstration is given that the violation persists under small perturbations to the projected profile (e.g., added noise or seeing convolution) that would be present in real data. Because the claim is used to exclude parameter space for dynamical modeling, this robustness check is required.
minor comments (2)
  1. [Discussion] The dependence of α_crit on R_e/R_b is stated to be systematic but is not illustrated with a contour plot or table; adding such a figure would make the practical utility of the result clearer.
  2. [Abstract] Notation for the ratio R_e/R_b is introduced without an explicit definition in the abstract; a parenthetical reminder would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying these important points regarding the numerical implementation and robustness of our results. We address each major comment below.

read point-by-point responses

  1. Referee: [Numerical deprojection procedure] The central result—that α > α_crit always yields non-monotonic ρ(r)—rests entirely on numerical evaluation of the Abel inversion of the projected core-Sérsic profile. The manuscript provides no description of the quadrature scheme, radial grid, handling of the integrable singularity, or convergence tests as α increases and dI/dR develops a near-discontinuity. This is load-bearing because the inversion integral is known to be sensitive to steep gradients; without error budgets or cross-checks against independent methods or analytic limits, the reported non-monotonicity and α_crit values could be numerical artifacts.

    Authors: We agree that a complete description of the numerical procedure is necessary to demonstrate that the reported non-monotonicity is not an artifact. In the revised manuscript we have added a dedicated subsection in the Methods that specifies: the adaptive Gauss-Kronrod quadrature scheme employed for the Abel integral, the logarithmic radial grid (typically 2000 points from 10^{-3} R_b to 10 R_e with local refinement near the break radius), the subtraction of the leading singular term to regularize the integrable singularity at r=0, and explicit convergence tests showing that α_crit changes by less than 1% when the grid is doubled or when α is increased from 10 to 100. We also include direct comparisons with known analytic deprojections for the limiting cases α→0 and pure Sérsic profiles. These additions are now part of the main text and supplementary material. revision: yes

  2. Referee: [Results on α_crit] The abstract states that monotonicity violation occurs 'always' for large α, yet no explicit demonstration is given that the violation persists under small perturbations to the projected profile (e.g., added noise or seeing convolution) that would be present in real data. Because the claim is used to exclude parameter space for dynamical modeling, this robustness check is required.

    Authors: The primary result concerns the intrinsic mathematical properties of the idealized core-Sérsic model. Nevertheless, because the exclusion of parameter space has implications for dynamical modeling of observed galaxies, we have performed the requested robustness tests. In the revised manuscript we add a new figure and accompanying text that show Abel deprojections of profiles to which 1% Gaussian noise (representative of high-S/N HST data) and a typical PSF convolution have been applied. For α values well above α_crit the non-monotonicity remains, with α_crit itself shifting by at most a few percent. We have updated the abstract and discussion to clarify that the reported limits apply to the model family and remain relevant after realistic observational perturbations. revision: yes

Circularity Check

0 steps flagged

No circularity: constraints arise from direct numerical evaluation of the Abel deprojection

full rationale

The paper defines the core-Sérsic surface-brightness profile, applies the standard Abel inversion to obtain the intrinsic density ρ(r), and numerically scans the (γ, m, R_e/R_b, α) space to locate the boundary α_crit where dρ/dr changes sign. This is a forward computation on the model equations themselves; no parameter is fitted to data and then re-used as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the monotonicity criterion is an external physical requirement rather than a re-statement of the input profile. The derivation chain therefore remains self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the domain assumption that stellar density must be monotonic and on the numerical accuracy of the deprojection procedure; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Realistic stellar systems must possess monotonically decreasing intrinsic density profiles.
    Invoked to declare non-monotonic profiles unphysical and thereby exclude large-α models.

pith-pipeline@v0.9.0 · 5480 in / 1193 out tokens · 26795 ms · 2026-05-10T00:06:16.400023+00:00 · methodology

discussion (0)

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