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REVIEW 4 major objections 4 minor 194 references

This paper claims that Balmer-line absorption occurs in roughly 35% of little red dots — about 850 times more often than in low-redshift type 1 active galactic nuclei — and that most of the absorbing gas is moving outward at modest speeds.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 00:19 UTC pith:27IFAECK

load-bearing objection A genuinely useful uniform Balmer-absorption census for LRDs; the qualitative contrast with type 1 AGNs holds, but the factor-850 headline is an observed ratio with an untested SDSS denominator. the 4 major comments →

arxiv 2607.26269 v1 pith:27IFAECK submitted 2026-07-28 astro-ph.GA

ATLAS. II. Extremely High Incidence of Balmer Line Absorption with Predominant Blueshifts in LRDs: Statistical Insights through Comparison with Type 1 AGNs

Hiroto Yanagisawa , Masami Ouchi , Tomokazu Kiyota , Makoto Ando , Yuichi Harikane , Yuta Kageura , Minami Nakane , Yoshiaki Ono
show 1 more author
Yui Takeda
This is my paper
classification astro-ph.GA
keywords little red dotsBalmer absorptionactive galactic nucleibroad-line regiondense gasoutflowsradiation pressuren=2 hydrogen
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper sets out to show that little red dots — compact, high-redshift galaxies with V-shaped ultraviolet-to-optical spectra — are systematically wrapped in dense, excited hydrogen. Using archival JWST/NIRSpec spectra of 40 such objects, the authors detect H-alpha absorption in 14 of them (35%), a rate roughly 850 times higher than in low-redshift type 1 active galactic nuclei, whose broad emission lines are directly visible. Combining their detections with published cases gives 46 absorption components, of which 83% are blueshifted relative to the [O III] systemic frame, with velocities confined to about -430 to +140 km/s. The paper argues that this narrow, mostly outward-moving pattern cannot be explained by detection incompleteness and that high-column absorbers (near 10^24 cm^-2) would remain gravitationally bound under single-scattering radiation pressure, implying stalled or recycling dense gas. The authors themselves flag that the literature compilation is heterogeneous and that the measured incidence is an observed lower limit; those caveats bound the statistics but do not erase the contrast with ordinary type 1 AGNs.

Core claim

The central discovery is that Balmer absorption — absorption of H-alpha and H-beta by hydrogen atoms in the n=2 excited state — is a common, statistically measurable feature of little red dots, not a rare curiosity: 14 of 40 (35%) in the uniformly selected sample show it, compared with 6 of 14,583 (about 0.04%) of low-redshift type 1 AGNs, a factor of roughly 850. In the combined sample of 46 absorption components, 38 (83%) are blueshifted relative to the [O III] lambda5007 systemic frame, with a median offset of -95 km/s and a range from -425 to +137 km/s; simulations injecting empirical AGN absorption profiles into the LRD spectra show that the object-dependent selection function cannot co

What carries the argument

The load-bearing object is the Balmer absorption line itself: H-alpha and H-beta absorption produced by hydrogen in the n=2 level along the line of sight to the continuum and broad-line region. The paper models it as a multiplicative partial-covering transmission T(lambda) = 1 - C_f + C_f exp[-tau(v)], with free optical depth, width, velocity centroid, and covering fraction, and adopts [O III] lambda5007 as the systemic-redshift frame. The analytic escape criterion E = (Delta-v_abs / v_esc)^2 + Gamma_eff > 1 ties the observed kinematics to dynamics: the first term is the measured absorber speed normalized by the gravitational escape velocity at the BLR radius, and Gamma_eff = lambda_Edd / (s

Load-bearing premise

[O III] lambda5007 gives an unbiased systemic redshift for the absorber velocities; if the narrow-line gas in little red dots is itself systematically moving (or blended with the Balmer-break continuum), then the measured blueshift fraction and the narrow -430 to +140 km/s velocity range could be artifacts of the chosen frame.

What would settle it

Re-measure the 46 absorbers using a systemic redshift independent of the nuclear narrow-line gas (for instance, stellar absorption or a spatially resolved low-ionization line) and re-derive the velocity distribution; if the blueshift fraction drops well below 83% or the velocity range widens beyond the reported -430 to +140 km/s, the kinematic pattern is a frame artifact. Conversely, detect Balmer absorption in a large sample of low-redshift type 1 AGNs using JWST-resolution spectra; an incidence approaching the LRD rate would mean the 850-fold contrast is mostly a resolution or sensitivity ef

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Balmer absorption becomes a population-level signpost of dense nuclear gas in little red dots, allowing covering fractions and kinematics to be measured statistically rather than object by object.
  • The roughly 850-fold incidence contrast implies the dense, n=2-populated hydrogen has a much larger effective covering fraction around LRD nuclei than around ordinary type 1 AGNs, supporting the dense-envelope picture.
  • The narrow, predominantly blueshifted velocity range (-430 to +140 km/s) indicates the absorbing gas moves slowly compared with BLR escape speeds; at columns near 10^24 cm^-2 most of it is bound and may stall and recycle.
  • The escape criterion identifies a column-density transition near 10^23-10^24 cm^-2 where the same outward-moving absorber changes from bound/failed to unbound/escaping; future column-density measurements will place individual absorbers on one side.
  • If the one-sided H-alpha-blue/H-beta-red centroid pattern is confirmed with more objects, it would point to a stratified structure with an inner infalling, higher-column component and an outer outflowing, lower-column component.

Where Pith is reading between the lines

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

  • A natural testable extension: apply the same incidence measurement to lower-redshift type 1 AGNs observed at NIRSpec-like resolution and signal-to-noise; if the contrast shrinks, resolution or sensitivity rather than intrinsic covering fraction may be driving the factor of 850.
  • If the [O III] systemic frame is locally biased by the dense-gas kinematics (for example, the narrow-line gas sharing the outflow), the 83%-blueshifted statistic could be partly frame-induced; comparing with stellar-absorption or spatially offset low-ionization line redshifts for a subset would settle this.
  • The bound-versus-escaped column threshold suggests a duty-cycle interpretation: the dense gas may be long-lived and recycling, in which case repeated spectroscopy on rest-frame timescales of months to years should show absorption profiles that vary in strength and velocity as clouds decelerate or are accelerated.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper presents a JWST/NIRSpec search for Balmer-line absorption in 40 color- and compactness-selected little red dots (LRDs) at z~2.5-7.2. It reports a ~35% incidence (14/40), claims this is ~850 times higher than in SDSS low-redshift type 1 AGNs (6/14,583), and combines the 14 new detections with 32 literature absorbers to build a 46-component census. From this census it reports that 83% of absorbers are blueshifted, that the velocity range (-430 to +140 km/s) is much narrower than in SDSS type 1 AGNs, and that, under a single-scattering radiation-pressure model, most absorbers with N_H ~ 10^24 cm^-2 remain gravitationally bound. The paper is explicitly framed as an observed-incidence lower limit on the LRD side, with extensive comparison to prior work.

Significance. If the central incidence claim holds, the paper provides a striking and quantitative demonstration that dense, excited hydrogen covers the nuclei of LRDs far more commonly than in ordinary type 1 AGNs, lending strong support to the 'dense envelope' interpretation. The compilation of 46 absorbers with transparent tables, the injection-recovery experiment for the velocity comparison, and the explicit acknowledgment of incompleteness are genuine strengths. The observed LRD incidence is consistent with the independent completeness-corrected estimate of 44+21/-6% cited from Juodzbalis et al. (2026), which is reassuring. However, the headline 'approximately 850 times' contrast is an observed ratio whose SDSS denominator completeness is not tested, and one secondary correlation is partly structural in the fitting model. These issues affect the strength of the population-level claims but are addressable.

major comments (4)
  1. [Section 4.1 / Figure 5] The central 850x contrast compares the observed LRD fraction (14/40) with the observed SDSS fraction (6/14,583). The paper correctly labels the LRD side a lower limit but does not test completeness on the SDSS side. The only argument given is resolution (R~1000 vs R~2000), but detectability also depends on S/N; weak, narrow Balmer absorption in heterogeneous SDSS spectra could be missed even at R~2000. If SDSS completeness were, say, 5%, the intrinsic contrast would drop by an order of magnitude. Please run an injection-recovery experiment on the SDSS spectra, or otherwise quantify SDSS completeness. If that is not feasible, the abstract and Section 5.1 should state that the 'approximately 850 times' figure is an observed-incidence ratio with unknown SDSS completeness, not a demonstrated physical contrast.
  2. [Section 2, compactness criterion] JADES-GN-68797 is the sole exception to the C444>0.5 compactness cut and is retained because it 'exhibits unambiguous Hα absorption.' This is a selection-on-the-outcome violation in the sample that directly feeds the numerator of the headline incidence. The numerical difference is small (14/40 vs 13/40), but the principle matters for a claimed statistical incidence. Please report the incidence with and without this object, and discuss whether other objects would be included if the compactness cut were relaxed. At minimum, quantify the sensitivity of the 'approximately 850x' factor to this single-object decision.
  3. [Section 4.3 / Eq. (2)] The strong correlation between EW_Hα,abs and EW_Hα,broad is partly structural. In Eq. (2), the transmission multiplies the continuum plus broad-Hα component; for fixed τ0, C_f, and σ_abs, the absorbed flux (and hence EW_abs) is proportional to the underlying broad-line flux. The paper acknowledges the middle panel is a 'recasting,' but the left panel is equally affected: a stronger broad line automatically yields a larger absorption equivalent width even if the absorber properties are unchanged. Please include a null test: inject a fixed absorption profile into mock spectra with varying EW_broad and show whether the observed correlation is reproduced. Absent that, the interpretation that the correlation indicates a 'common increase in the amount and geometrical covering of dense gas' is not yet supported.
  4. [Section 3, systemic-redshift frame] All velocity statistics (83% blueshifted, the -430 to +140 km/s range, the v_esc normalization, and the escape diagram) are measured relative to [O III] λ5007. If [O III] itself is not a faithful systemic tracer (e.g., the narrow-line gas participates in the same dense kinematics, or the line is blended with the Balmer-break continuum inflection), the inferred predominance of blueshifts and the narrow velocity range could be biased. The paper notes the compiled census is heterogeneous in systemic tracer, but the new measurements all use [O III]. Please assess the sensitivity by cross-checking with alternative tracers where available (e.g., [O II], stellar absorption, other narrow lines) or by quantifying a plausible systematic shift. This does not affect the incidence claim, but it is load-bearing for the kinematic and escape conclusions.
minor comments (4)
  1. [Abstract / Section 5.3] Grammar: 'most absorbers with N_H > 10^24 cm^-2 remains gravitationally bound' should be 'remain.' Also, the phrase 'The smaller number of redshifted (i.e., infalling) absorbers' is loose: redshifted absorption is only infalling if the systemic frame is correct; see major comment on [O III].
  2. [Appendix Tables 1-2] The tables are extremely useful and comprehensive, but the many repeated entries make them hard to read. Consider visually separating the 'this work' measurements from the literature compilation, and adding a column that flags which entries are used in each population statistic (e.g., median for repeated epochs, unique components).
  3. [Eq. (3)] Please clarify that FWHM_Hα,broad is in velocity units and specify whether the instrumental resolution correction is applied before taking the square root. The text says 'we use the FWHM of the total broad-Hα model profile including the electron-scattering wings,' which is clear, but the equation alone could be misread.
  4. [Figure 9 caption] The caption states 'the three lower-S/N objects are shown but excluded.' Please list the three object names, since the reader cannot identify them from the figure alone.

Circularity Check

1 steps flagged

The headline ~850x incidence claim is not circular; the main circularity is a secondary EW_abs vs EW_broad correlation that is partly built into the multiplicative absorption model (Eq. 2), with the paper itself conceding the middle panel is a recasting.

specific steps
  1. self definitional [Section 3, Eq. (2) and Section 4.3, Figure 9]
    "we model the absorption as a multiplicative transmission applied to the continuum plus broad Hα component: T(λ) = 1−C_f +C_f exp[−τ(λ)] ... From the best-fit transmission profile, we measure the rest-frame equivalent width of the absorption as EW_abs = ∫[1−T(λ)]dλ/(1+z) ... Note that these two quantities are not independent because both express the broad-Hα line strength relative to the optical continuum; the middle panel should therefore be regarded as a recasting."

    Because the transmission T(λ) multiplies the continuum plus the broad-Hα component, EW_abs is, at fixed absorber parameters, approximately C_f(1−e^{−τ}) [Δλ + EW_broad] (or, equivalently, affine in L_Hα,broad/L_5100). Therefore a strong Spearman correlation between EW_abs and EW_broad is largely guaranteed by the fitting construction, not by an independent physical coupling. The paper reports ρ=0.917 and interprets it as a 'common increase in the amount and geometrical covering of dense gas,' but concedes only that the middle panel is a recasting; the same structural dependence applies to the left panel because the absorption is applied multiplicatively to the broad line. This is a partial circularity in a secondary result, not in the headline incidence claim.

full rationale

The central incidence claim—14/40 = 35% in LRDs vs 6/14583 ≈ 0.04% in SDSS type 1 AGNs, i.e. ~850x—is not circular: the numerator and denominator on both sides are directly measured, the LRD fraction is explicitly treated as a lower limit, and the SDSS comparison is an external sample (Shangguan et al. 2026). The velocity-distribution result is tested with an injection–recovery forward model using the observed LRD spectra and empirical SDSS profiles, so it is not manufactured by the fitting procedure. The v_esc normalization and radiation-pressure escape criterion are standard analytic scalings, not fitted outputs. The genuine structural circularity is in Section 4.3: since the absorption model multiplies the continuum plus broad-Hα flux (Eq. 2) and EW_abs is measured from that same transmission, EW_abs is approximately affine in EW_broad at fixed absorber properties; the reported strong correlations are therefore partly built into the model. The paper explicitly concedes that the middle-panel quantity is a recasting but still uses the left-panel correlation as evidence for a common physical increase in dense gas, and the interpretation also leans on the authors' own ATLAS I paper. This is secondary rather than load-bearing for the headline result, so score 4 rather than 6+. The skeptical concern about untested SDSS completeness is a correctness risk, not a circularity, and does not raise the circularity score.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central claims rest on a public archival sample (DJA v4.4), hand-set color/compactness cuts that define the population, literature-calibrated virial and bolometric constants, and an unconstrained N_H whose assumed range controls the escape conclusion. No new physical entities are introduced; the 'failed flow/recycling' language describes a scenario, not a new entity.

free parameters (6)
  • LRD color thresholds (Eq. 1) = z<4: F150W-F277W>1.0, F090W-F115W<1.0; 4<z<6: F200W-F356W>1.0, F115W-F150W<1.0; z>6: F277W-F444W>1.0, F150W-F200W<1.0
    Hand-chosen per-redshift-bin cuts; the authors state 'these selections are arbitrary to some extent' (Section 2). They define the population whose incidence is 14/40.
  • Compactness threshold C444 = 0.5
    Adopted from Akins et al. 2025; JADES-GN-68797 is retained below it (C444=0.398) because it shows the signal of interest (Section 2), directly affecting the 14/40 numerator.
  • Virial factor f = 1.12
    Adopted from Woo et al. 2015; sets v_esc = sqrt(2f) FWHM_Halpha,broad (Eq. 3), used for the velocity normalization and the escape criterion.
  • Bolometric correction L_bol = 19 L_Halpha,broad = 19
    Adopted from Greene et al. 2025; converts Halpha luminosity into lambda_Edd for Eq. (5). A wrong factor shifts N_H,crit linearly; median lambda_Edd is 0.10 (observed widths) or 0.25 (intrinsic widths).
  • Total hydrogen column N_H = free; 10^23-10^26 cm^-2 scanned, fiducial 10^24 cm^-2
    Unconstrained in the escape diagram (Fig. 11); the paper states individual fates are 'not established' (Section 5.3). The bound-versus-escaping headline statement is a function of this parameter.
  • Per-object absorption parameters (v_abs, sigma_abs, tau_0, C_f) = Table 2 values
    Four free parameters per object in Eq. (2), selected via Delta-BIC > 10 plus visual inspection (Section 3); all velocity and EW results derive from these fits.
axioms (6)
  • domain assumption [O III] lambda5007 is an unbiased systemic-redshift tracer for the absorber velocity frame
    All Delta-v_abs measurements are relative to [O III] (Section 3); the 83%-blueshifted statistic and the narrow -430 to +140 km/s range depend on this frame. The compiled 46-absorber sample mixes different systemic tracers, acknowledged in Section 3.
  • domain assumption The SDSS Balmer-absorption profiles (Shangguan et al. 2026) are a representative template library for LRD absorbers
    The injection-recovery conclusion (Section 4.2) that incompleteness cannot explain the velocity difference assumes intrinsic LRD absorber profiles resemble the empirical SDSS ones; different optical-depth structure could alter recovery statistics.
  • domain assumption The LRD (40 spectra) and SDSS (14,583 spectra) incidence denominators are comparably sensitive per sightline modulo resolution
    The 850x contrast (Section 4.1) treats the two surveys as measuring the same physical quantity. LRD-side incompleteness is acknowledged (it would raise the contrast), but SDSS-side completeness for shallow Balmer absorption is not quantified.
  • domain assumption Single-scattering complete momentum coupling: Gamma_eff ~ lambda_Edd/(sigma_T N_H)
    Stated as an 'idealized limit' in Section 5.3; it maximizes radiative acceleration at fixed bolometric luminosity. The E>1 escape boundary (Eq. 5) and N_H,crit (Eq. 6) rest on it.
  • standard math Virial relation v_esc = sqrt(2f) FWHM_Halpha,broad with f = 1.12
    Eq. (3) converts the fitted broad-Halpha FWHM into an escape velocity at the BLR radius, assuming the virial/BLR velocity field and the adopted virial factor; the R_BLR-L relation cancels but the FWHM-to-velocity relation is assumed.
  • standard math Planck 2020 cosmology
    Adopted for distance and luminosity conversions (stated in Section 1).

pith-pipeline@v1.3.0-alltime-deepseek · 28815 in / 23080 out tokens · 218538 ms · 2026-08-01T00:19:21.141897+00:00 · methodology

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read the original abstract

We present the statistical properties of H$\alpha$ and H$\beta$ line absorption in little red dots (LRDs) at $z\simeq2.5$--7.2 using archival JWST/NIRSpec spectra from the DAWN JWST Archive and complementary NIRSpec/IFU observations. Among 40 LRDs with broad H$\alpha$ and [O~{\sc iii}] obtained with medium- or high-resolution gratings, 14 objects exhibit H$\alpha$ absorption. We find that the incidence of Balmer line absorption is $\sim35$\% ($=14/40$), significantly higher than that in SDSS low-$z$ type 1 AGNs ($\sim0.04$\%), demonstrating that Balmer line absorption occurs approximately 850 times more frequently in LRDs than in type 1 AGNs. We combine our 14 detections with 32 additional LRD Balmer absorbers from the literature, yielding a census of 46 absorbers. Their velocities span $\Delta v_\mathrm{abs}(\mathrm{H\alpha})\sim-430$ to $+140\ {\rm km\,s^{-1}}$, markedly narrower than the $-800$ to $+1600\ {\rm km\,s^{-1}}$ range of Balmer absorption in SDSS type~1 AGNs, for which our simulations confirm that the velocity difference is too large to be explained by detection incompleteness. The lower absolute absorber velocities in LRDs may partly reflect the shallower gravitational potential at their characteristic BLR radii. We also find that 38 of the 46 absorbers (83\%) are blueshifted, with only eight redshifted, indicating that most of Balmer absorbers are moving outward. An analytic model with radiation pressure suggests that most absorbers with $N_{\rm H}\gtrsim10^{24}\ {\rm cm^{-2}}$ remains gravitationally bound. The smaller number of redshifted (i.e., infalling) absorbers may indicate that outbound absorbers lose density: some return to the BLR, whereas others undergo stronger radiative acceleration and escape.

Figures

Figures reproduced from arXiv: 2607.26269 by Hiroto Yanagisawa, Makoto Ando, Masami Ouchi, Minami Nakane, Tomokazu Kiyota, Yoshiaki Ono, Yuichi Harikane, Yui Takeda, Yuta Kageura.

Figure 1
Figure 1. Figure 1: LRD color criteria presented in Equation (1). From left to right, the panels show the criteria for z < 4, 4 < z < 6, and z > 6. The pink shaded regions indicate the adopted color-selection windows. Filled red circles show adopted LRDs with Hα absorption. Open gray circles show the remaining broad-Hα sources in the parent sample. fscatt of which is redistributed into exponential wings by electron scattering… view at source ↗
Figure 2
Figure 2. Figure 2: JWST/NIRSpec medium- or high-resolution grating spectra of the 14 LRDs with detected Hα absorption. For each object the Hβ panel (left) is shown together with the Hα panel (right) on a common rest-wavelength scale. Black lines show the observed spectra, gray shaded regions show the 1σ uncertainties, red curves show the best-fit line-profile models, red dashed vertical lines mark Hα and Hβ at the systemic r… view at source ↗
Figure 3
Figure 3. Figure 3: Same as [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Balmer-absorption fraction for LRDs and low-z type 1 AGNs. Red circles show the LRD sample in this work, and black squares show the SDSS type 1 AGN comparison sample constructed from J. Shangguan et al. (2026). Er￾ror bars show 95% Wilson confidence intervals; downward arrows denote one-sided upper limits for bins with zero de￾tected absorbers. The solid red and black lines show the observed overall fracti… view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of the Hα absorption velocity offset. (a) Measurements separated by study. The filled red histogram shows this work; orange frequency polygons with distinct markers show the bin counts from A. J. Taylor et al. (2025), J. Matthee et al. (2026), K. Davis et al. (2026), C.-H. Chen et al. (2026), I. Juodˇzbalis et al. (2026), X. Lin et al. (2026), and K. Park et al. (2026); and the hatched gray hi… view at source ↗
Figure 7
Figure 7. Figure 7: Forward modeling of the Hα absorption velocity distribution. The histogram shows the injected empirical SDSS absorption-profile (hatched gray histogram), the mock observed distribution (blue outline), and the observed LRD absorbers (red histogram). matic information at different cloud depth. The absorp￾tion profiles are superposed on different combinations of narrow and broad emission between Hα and Hβ, ma… view at source ↗
Figure 8
Figure 8. Figure 8: Hβ versus Hα absorption velocity offsets. Red cir￾cles show this work; orange symbols show literature measure￾ments from I. Juodˇzbalis et al. (2026), X. Lin et al. (2026), K. Davis et al. (2026), E. Lambrides et al. (2025), A. Tor￾ralba et al. (2026a), L. R. Ivey et al. (2026), F. D’Eugenio et al. (2026b), X. Ji et al. (2026), X. Lin et al. (2025), and J. Matthee et al. (2026). All 32 measurements from 11… view at source ↗
Figure 9
Figure 9. Figure 9: Rest-frame Hα absorption equivalent width, EWHα,abs, as a function of the rest-frame broad-Hα emission equivalent width (left), the broad-Hα-to-5100 ˚A continuum luminosity ratio, log(LHα,broad/L5100) (middle), and the signal-to-noise ratio of the broad Hα component (right). Only the 12 absorbers with an available L5100 measurement are shown. Red circles denote the nine LRDs with S/NHα,broad > 20 used in t… view at source ↗
Figure 10
Figure 10. Figure 10: Absolute velocity offset of the Hα absorbers, |∆vabs(Hα)|, as a function of the escape velocity at the BLR radius, vesc = √ 2f FWHM (Equation 3). Only blueshifted (outflowing) absorbers are shown. Red filled circles show the LRD absorbers with vesc computed from the observed FWHM including the electron-scattering wings, while red open circles show the same objects with vesc from the intrin￾sic Gaussian FW… view at source ↗
Figure 11
Figure 11. Figure 11: Escape diagram based on the conditional crite￾rion of Equation (5). Only blueshifted (outflowing) absorbers are shown. The black curve marks E = 1, and the yellow shaded region is unbound within the idealized fixed-column, radial, point-mass model. Symbols are the same as [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗

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