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Redefining $Q$ for multi-component discs of stars and gas

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A redefined disc-stability Q maps one-to-one to spiral amplification, and raises the Solar Neighbourhood value to 1.58.

desk verdict Gas Q should indeed be roughly squared for swing amplification, but the calibration and validation both lean on the same GLB model, so treat δ=5/3 as a useful internal consistency result until independent tests appear. read the letter →

arxiv 2502.05304 v1 pith:PJWERZUF submitted 2025-02-07 astro-ph.GA

Kit George , Ralph Schönrich This is my paper · ORCID
classification astro-ph.GA
keywords ToomreQswingamplificationdiscstabilitymulti-componentdiscsinterstellargasspiralstructureMilkyWayM74
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

Toomre's Q is the standard stability parameter for galactic discs, but it was built to mark the boundary of axisymmetric instability at $Q=1$, not to measure spiral activity above that boundary. The paper argues that using $Q$ at $Q>1$ as a gauge of spiral strength is nevertheless common practice and, for gas-rich discs, numerically wrong. Working from the Goldreich-Lynden-Bell swing-amplification formalism, the authors show that the gas $Q$ should be redefined close to the square of its traditional value, and they construct a multi-component parameter $Q_{\delta=5/3}$ that gives a close one-to-one map between $Q$ and the maximum swing amplification factor while preserving $Q=1$ as the axisymmetric stability boundary. Applied to the Solar Neighbourhood the new definition raises $Q$ from about 1.44 to 1.58, closer to simulation values, and it predicts that high-$m$ gas-dominated instabilities, not stellar spiral arms, produce the flocculent gas structure seen in the Milky Way and M74.

What carries the argument

The load-bearing machinery is the Goldreich-Lynden-Bell (GLB) swing-amplification calculation, extended to discs with multiple stellar and gas components through Rafikov's WKB dispersion relation. The GLB equation gives the reduced spring rate of a shearing density wave as a function of wave inclination; when this rate is negative the wave grows, and integrating the growth over the swing from leading to trailing gives a maximum amplification factor. The paper uses this factor as the target: a useful $Q$ must make maximum amplification a decreasing, one-to-one function of $Q$ (requirement (ii) in Section 2.2). The new parameter $Q_\delta$ (equation 64) is defined by a fixed-point condition on the dispersion relation—multiply all surface densities by $Q_\delta$, scale gas sound speeds by $Q_\delta^{1-1/\delta}$, and require marginal axisymmetric stability—so that the map between $Q$ and amplification is as composition-independent as possible. The choice $\delta=5/3$ is the value that minimizes the spread $\Delta_{\mathrm{GLB}}$ across gas fractions and sound-speed ratios.

What would settle it

Run a hydro/N-body simulation that holds $Q_{\delta=5/3}$ fixed while changing the gas fraction from about 0.25 to 0.75; if the measured spiral amplitude differs substantially between the runs, the claimed one-to-one map between $Q$ and swing amplification fails. A second test is observational: resolved cold-gas maps of the Milky Way should show recurrent $m \approx 10$–$25$ spirals independent of the stellar arms if the predicted gas instability is real.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that $Q$ as traditionally defined conflates two different instabilities: the axisymmetric stability boundary ($Q=1$) and the susceptibility to non-axisymmetric spiral growth via swing amplification (the regime $Q>1$). While this conflation is harmless for a single stellar population, it becomes a real numerical error when gas is present. Using the Goldreich-Lynden-Bell shearing-sheet calculation extended to multi-component discs through the Rafikov WKB dispersion relation, the authors find that the standard gas $Q$ must be redefined close to its square to put gas and stellar discs on the same amplification scale. They then define a new multi-component parameter $Q_\delta$ (equation 64, optimum $\delta = 5/3$) by demanding that multiplying all surface densities by $Q_\delta$ while scaling the gas sound speed by $Q_\delta^{1-1/\delta}$ brings the disc to marginal axisymmetric stability. This parameter preserves $Q=1$ as the stability boundary, reduces to the standard stellar $Q$ for a pure stellar disc, and in the authors' tests gives a close one-to-one relationship between $Q$ and maximum swing amplification across gas fractions. The paper concludes that $Q_\delta$ should replace $Q_{\mathrm{Rk}}$ as the standard stability measure for thin multi-component discs.

Load-bearing premise

The load-bearing premise is that the maximum swing-amplification factor from the local Goldreich-Lynden-Bell shearing-sheet model, extended to stars and gas through Rafikov's dispersion relation, is the right measure of spiral activity, so that minimizing the composition-dependent scatter in that factor is the correct design criterion for a new Q; if that model does not track real spiral growth, the proposed Q is merely a re-parametrisation of an approximate calculation.

Editorial extensions

If this is right

  • For any disc containing gas, the traditional gas $Q$ must be redefined close to its square, so gas-rich discs are more stable at a given classical $Q$ than previously assumed, changing the relative weight of stellar and gas contributions to spiral activity.
  • The new parameter $Q_{\delta=5/3}$ (equation 64) is recommended as the standard stability measure for thin multi-component discs, replacing $Q_{\mathrm{Rk}}$; an iterative formula (equation 65) makes it straightforward to compute.
  • The Solar Neighbourhood's $Q$ rises from about 1.44 under the Rafikov definition to 1.58 under $Q_{\delta=5/3}$, bringing the local disc estimate into line with self-regulated N-body simulations that settle near $Q \approx 1.5$.
  • Swing amplification peaks broadly in azimuthal wavenumber $m$: stellar-dominated patterns favour low $m$, while gas-dominated instability appears at $m \gtrsim 10$, which the paper links to the flocculent gas structure in M74 and local spurs in the Milky Way.
  • For the Milky Way, $m=2$ waves have maximum amplification below 1.25, too weak to sustain a two-armed pattern through feedback, whereas $m=4$ waves are viable; M74's $m=2$ grand design is close to its amplification peak.

Reading between the lines

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

  • Beyond the paper: if the gas $Q$ is indeed squared, published comparisons of stability between gas-rich and gas-poor galaxies using the classical gas $Q$ may have systematically underestimated the stability of gas-rich systems; re-analysis of existing $Q$ catalogues would reveal how large the correction is in practice.
  • Beyond the paper: a direct controlled N-body/hydro experiment that fixes $Q_{\delta=5/3}$ and varies gas fraction would test the one-to-one claim more cleanly than the paper's existing simulation comparison, which is not designed for this purpose.
  • Beyond the paper: the same fixed-point construction that defines $Q_\delta$ could be applied to multi-age stellar populations, providing a composition-independent stability parameter for discs with arbitrary stellar age distributions, an extension the paper only sketches.
  • Beyond the paper: if the gas-instability link is right, high-resolution cold-gas surveys of the Milky Way should reveal ubiquitous $m \approx 10$–$25$ spiral structure independent of stellar arms, which can be checked against existing molecular-line observations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper argues that Toomre's Q conflates two distinct notions: the axisymmetric stability boundary at Q=1 and the susceptibility to spiral activity for Q>1. It proposes redefinitions that preserve Q=1 but give a more nearly one-to-one map between Q and the maximum swing-amplification factor computed from the Goldreich-Lynden-Bell (GLB) shearing-sheet formalism. The first proposal, Q_dr (Eq. 52), equals the square of the classical gas Q and is close to the classical stellar Q. A refined hybrid parameter Q_delta (Eq. 64) is then introduced, with an exponent delta=5/3 chosen by minimizing the composition scatter of the GLB-based measure Delta_GLB (Eq. 46). The paper provides an iterative formula and code, applies the new parameter to the Solar Neighbourhood (Q≈1.58) and to M74, and predicts that high-m, gas-dominated instabilities produce flocculent ISM structure and Milky Way spurs.

Significance. If the proposed redefinition is correct, it would resolve a known inconsistency in the use of Q for star-gas discs and would provide a practical replacement for Q_Rk, with quantitative consequences for Milky Way and M74 spiral structure. The paper contains a clean analytic identity (Eqs. 50-52) showing Q_dr=Q_g^2 for a single gas disc, a transparent numerical comparison of several Q definitions, and publicly available code for the iterative computation. The authors are also candid about the limitations of the GLB formalism and about the lack of error propagation in the applications. However, the headline recommendation Q_delta=5/3 is calibrated and validated with the same approximate GLB model that defines the criterion, and no independent test connects the resulting parameter to real spiral amplitudes in simulations or observations. The significance is therefore real but conditional on an external validation that the current manuscript does not provide.

major comments (3)
  1. [Sec. 4.2, Eq. (64), Fig. 10] The exponent delta=5/3 is selected by minimizing the spread of Delta_GLB (Eq. 46), and the same Delta_GLB is then used to demonstrate the improvement of Q_delta over previous definitions. This is an internal-consistency calibration, not an independent validation. Since Sections 1 and 3 state that the GLB calculation 'cannot be used to describe the evolution of real spiral patterns' and is only a 'useful indicative measure', the central claim that Q_delta=5/3 is the 'correct' definition for spiral activity requires an external test, e.g. against N-body/gas simulations or against the Julian-Toomre formalism extended to gas-rich discs. Without such a test, the quantitative recommendation rests entirely on the GLB model the paper itself calls approximate.
  2. [Sec. 4.1, Figs. 8-9; requirement (ii) in Sec. 2.2] Requirement (ii) demands a decreasing one-to-one map between Q and maximum swing amplification, but Fig. 9 shows that the maximum amplification flattens for Q_dr≳3, and the text admits that flat gradients allow discs of different composition with very different Q to share the same maximum amplification. Thus the proposed definition fails its own one-to-one criterion in the high-Q regime. Moreover, the recommended Q_delta is never subjected to the Delta_JT test that is used for Q_dr in Fig. 8, so it is unknown whether Q_delta satisfies requirement (ii) for multi-component stellar discs. The paper should either demonstrate one-to-one behaviour for Q_delta over the range it recommends, or explicitly restrict the claimed validity to Q≲3.
  3. [Sec. 5.1, Eqs. (90)-(93)] The comparison with the Aumer et al. (2016) simulation value Q≈1.5 is not a strong validation because the authors state 'Without full error propagation' and vary only the gas surface density, ignoring the quoted uncertainties in kappa (41±2.5 km/s/kpc) and stellar surface density (38±4 M_sun/pc^2). The resulting range Q_delta=1.06-2.13 completely brackets both the new and old central values, so the claim that Q≈1.58 is 'closer to results from simulations' is not robust. A full error estimate should be provided, or the claim should be weakened accordingly.
minor comments (5)
  1. [Abstract and Sec. 6] The abstract and conclusion state that the gas Q should be redefined to 'close to the square' of the traditional definition, but the recommended Q_delta=5/3 (Eq. 64) reduces to Q_g^{5/3} for a pure gas disc, not Q_g^2; the Q^2 relation holds only for the intermediate Q_dr. The wording should be adjusted to match the final recommended definition.
  2. [Sec. 4.3, around Eq. (79)] The text 'epsilon between 11 6 and 2' contains a typographical artifact ('11 6' should be '11/6'). Since the recommended epsilon is stated as approximately 2, this is a presentation issue, but the broken fraction should be fixed.
  3. [Sec. 5.2, after Table 3] The choice of kappa=85 km/s/kpc to obtain Q_delta=1.03, after noting that the inferred value 81.6 gives Q_delta=0.99, is presented without uncertainty quantification. Since the galaxy is face-on and the rotation curve is poorly constrained, the statement that the disc is 'marginally unstable' or 'marginally stable' needs an error bar or a more careful caveat.
  4. [Sec. 3, Eq. (45)] The definition of X=k_crit/k_y and the sampling choices (250 values of X, 50 initial phases, 1000 timesteps) are given in the text, but there is no convergence test for these numerical parameters; a brief convergence statement would strengthen confidence in the reported amplification factors.
  5. [Data availability] The statement that code 'will be shared on reasonable request' is followed by a GitHub link; the link should be described as the canonical source and, ideally, the notebook should be archived with a version identifier to support reproducibility.

Circularity Check

1 steps flagged · score 6.0 of 10

Q_δ's one-to-one map to swing amplification is calibrated and validated with the same GLB model, so the headline improvement is partly by construction; external checks keep it from being fully circular.

  1. fitted input called prediction [Section 3.1, Eq. (46); Section 4.2, Fig. 10; Section 1 caveat]
    "We can define a relative error in Q from this ideal: ΔGLB(Σs/Σg, σR/c, Q0)=... (46) We wish to minimize |ΔGLB| across various values of Σs/Σg, σR/c and Q0, which is synonymous to approaching a one-to-one mapping between the Q value and amplification under each given definition. ... Figure 10 shows ΔGLB for two-component discs of stars and gas with various values of Σs/Σtotal and σR/c, for Qδ with multiple values of δ. δ ≈ 5/3 appears to give the closest to a one-to-one correspondence between Qδ and maximum amplification."

    The quality metric ΔGLB (Eq. 46) is defined as the deviation of a Q definition from a one-to-one map to the maximum GLB amplification. The exponent δ in Qδ is then selected by minimizing |ΔGLB| over a grid of disc compositions (Fig. 10). The subsequent demonstration that Qδ=5/3 has small scatter in ΔGLB relative to other definitions is a calibration check, not an independent test, because the objective and the validation metric are the same GLB amplification model. The paper itself states that the GLB calculation 'cannot be used to describe the evolution of real spiral patterns in terms of swing amplification' and calls it only 'a useful indicative measure' (Section 1).

full rationale

The paper's central new quantity Qδ is defined through Eq. (62)–(64) as a generalization of Rafikov's QRk, with a free exponent δ that controls how gas sound speeds are rescaled. The paper selects δ ≈ 5/3 by minimizing ΔGLB (Eq. 46), which measures the scatter in the maximum amplification factor computed with the same GLB shearing-sheet model that is later used to validate Qδ (Figs. 5–10). This is a genuine calibration loop: the objective function and the demonstration of the 'one-to-one map' are the same GLB calculation. The paper is honest about the approximate nature of GLB, calling it only 'a useful indicative measure' and noting it 'cannot be used to describe the evolution of real spiral patterns in terms of swing amplification' (Section 1). That caveat limits the external force of the calibration. However, several independent elements break the loop: (i) the gas Q → Q^2 relation (Eq. 51–52, Fig. 3) is analytic, derived directly from the single-fluid dispersion relation, and is not fitted to ΔGLB; (ii) the Solar Neighbourhood value Qδ ≈ 1.58 is compared to Aumer et al. (2016) simulations, an independent N-body benchmark; (iii) the Milky Way and M74 applications connect to observational spiral multiplicity and flocculence. These provide real, non-circular support for the qualitative claim that gas Q has been mismatched. The circular component is limited to the specific exponent δ≈5/3 and the claim that Qδ provides the best one-to-one map to swing amplification: that claim reduces, by construction, to optimizing a metric defined with the same GLB model. I therefore score 6, reflecting that a central 'prediction' (the one-to-one map) is partly an artifact of the fitting procedure, while the broader gas-square and the observational applications retain independent content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central construction rests on the GLB and JT swing-amplification formalism, the Rafikov multi-component dispersion relation, and the normative requirement that Q should be one-to-one with maximum amplification. The exponents delta = 5/3 and epsilon near 2 are fitted to the resulting Delta_GLB curves rather than derived, while the Solar Neighbourhood and M74 applications further assume literature input values for surface densities, velocity dispersions and epicycle frequencies. No new physical entities are introduced.

free parameters (2)
  • delta (exponent in Q_delta definition) = 5/3
    Chosen to minimize the spread in Delta_GLB across two-component discs with different star/gas mass ratios and dispersion ratios; not derived from first principles. Section 4.2 and Figure 10.
  • epsilon (vertical exponent in Q_delta,epsilon) = 2 (range 11/6 to 2)
    Selected from Delta_GLB plots for thick discs using approximate Romeo (1992) thickness corrections; the authors call the thickness treatment a curiosity due to problematic assumptions. Section 4.3 and Figures A1-A2.
assumptions (4)
  • domain assumption The maximum swing amplification factor from the GLB shearing-sheet calculation is the correct physical quantity for spiral instability, so Q should be calibrated to be one-to-one with it.
    This is the paper's stated design criterion, Section 2.2 requirement (ii), and the basis of the Delta_GLB metric, Eq. 46. If wrong, the redefinition has no physical motivation.
  • domain assumption The WKB dispersion relation remains a valid local response for the shearing waves, including near gamma near 0 where the waves are not tightly wound.
    The GLB equation is used with s^2(k) from the Rafikov (2001) dispersion relation, Section 3 Eqs. 38-39; the authors note the WKB condition fails near gamma = 0 and rely on similarity with the Julian-Toomre calculation.
  • standard math The Rafikov (2001) dispersion relation, Eq. 10, correctly describes a disc with multiple isothermal gas components and Schwarzschild stellar components.
    Taken as the starting point for all multi-component Q definitions; a cited published result used without derivation in this paper.
  • domain assumption The Romeo (1992) finite-thickness reduction factors are accurate enough to describe realistically thick discs.
    The authors explicitly caution that these corrections are based on problematic assumptions and present the thick-disc treatment only as a curiosity, yet it is used to define Q_delta,epsilon in Section 4.3.

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Pith. "Pith review of Redefining $Q$ for multi-component discs of stars and gas." pith.science (2026). https://pith.science/paper/PJWERZUF

@misc{pith2026250205304,
  author       = {Pith},
  title        = {Pith review of: Redefining $Q$ for multi-component discs of stars and gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJWERZUF}},
  note         = {Machine review of arXiv:2502.05304}
}
abstract

We point out a fundamental mismatch in the $Q$ stability parameter for Galactic discs: Toomre's $Q = 1$ defines the boundary between axisymmetric stability/instability, while simulations, observations, and theoretical expectations apply $Q$ in the region $Q > 1$ as a measure for spiral activity (e.g. swing amplification), for which $Q$ has not been designed. We suggest to redefine $Q$ to keep $Q = 1$ as the stability boundary, but to equally yield a consistent map between $Q$ and the maximum swing amplification factor. Using the Goldreich-Lynden-Bell formalism, we find that particularly the $Q$ for gas discs has been mismatched, and should be redefined to close to the square of the traditional definition. We provide new formulations of $Q$ for simple, two-component, and multi-component discs, including a discussion of vertically extended discs, providing a simple iterative formula for which we also provide code. We find $Q \approx 1.58$ for the Solar Neighbourhood under our definition, closer to results from simulations. We compare the Milky Way and M74, showing that, consistent with observations, the theory suggests a higher $m$ number for the Milky Way (arguing against a 2-arm pattern) for stellar-dominated patterns. Gas instability arises at much smaller scales ($m \gtrsim 10$), and we link both M74's gas pattern and local spurs in the Milky Way to this gas instability rather than stellar spiral arms.

Figures

Figures reproduced from arXiv: 2502.05304 by the authors.

Figure 1
Figure 1. Contours of constant 𝑄Rk (equation 23) for a 2-component disc of stars and gas with individual 1 𝑄 values for its gas and stellar components 𝑄g and 𝑄s respectively, for various mass ratios between the two components. Each contour is an enlargement of the 𝑄Rk = 1 contour about the origin by factor 𝑄Rk. Definition Notes 𝑄Rk = 𝐹max ( {Σg,𝑖 }, {Σs, 𝑗 }, {𝑐𝑖 }, {𝜎𝑅, 𝑗 } )−1 Best existing definition for a disc of stars an… view at source ↗
Figure 2
Figure 2. Maximum amplification (choosing optimal initial epicycle phase) achievable in stellar discs of various 1 𝑄 values, as a function of 𝑋 = 𝑘crit/𝑘𝑦 (azimuthal wavenumber 𝑘𝑦 = 𝑚/𝑅 for a spiral with multiplicity 𝑚), as calculated using the Goldreich-Lynden-bell (GLB) equation (equation 37 and the Julian-Toomre (JT) equation (equation 41) respectively. See similar plots in Toomre (1981) and Binney (2020). 1.0 1.5 2.0 2.5 … view at source ↗
Figure 3
Figure 3. Our first redefinition of 𝑄, 𝑄dr (equation 52), as a function of the respective standard definitions of 𝑄 for single component discs of gas or stars. Note that 𝑄dr = 𝑄2 for gas, but 𝑄dr ≈ 𝑄 for stars. If 𝑄dr is a consistent measure of spiral instability then this implies that the standard definition of 𝑄 is inconsistent in comparison between discs of stars and discs of gas when 𝑄 > 1, and that this inconsistency gro… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Existing multi-component definitions for 𝑄 as a function of our new definition 𝑄dr for two-component discs of stars and gas, for various mass ratios and sound-speed/velocity dispersion ratios between the two components. All definitions roughly agree with 𝑄dr for a pure…
Figure 5
Figure 5. Figure 5: Validation of 𝑄 via consistency in amplification: Maximum amplification (with respect to 1 𝑋 = 𝑘crit/𝑘𝑦 and initial epicycle phase) achievable in a two-component disc of stars and gas with various mass ratios between the two components and with gas sound speed and stel…
Figure 6
Figure 6. Figure 6: ΔGLB for a 2-component disc of stars and gas as a function of 𝑄0 for 𝑄dr and the existing multi-component 𝑄 definitions, for various mass ratios and sound speed/velocity dispersion ratios between the two components. Definitions are better, the closer they remain to 0 a…
Figure 7
Figure 7. Figure 7: 𝑄 1 Rk value of a many-component stellar disc with star formation rate history SFR(𝑡) ∝ exp(±𝑡/𝑡SFR) and age-velocity dispersion relation 𝜎𝑅 (𝜏) ∝ 𝜏 𝛽 as a function of 𝛽, for various SFRs and for various fixed values of 𝑄 when the disc is treated as a single component,…
Figure 8
Figure 8. Figure 8: Δ 1 JT (equation 47) as a function of 𝑄0 for 𝑄dr and the existing multi-component 𝑄 definitions, for a 100-component stellar disc with constant star formation rate (SFR) and with age-velocity dispersion relation (AVR) 𝜎(𝜏) ∝ 𝜏 𝛽 , for various values of 𝛽. For values of…
Figure 9
Figure 9. Figure 9: Maximum amplification (with respect to 𝑋 = 𝑘crit/𝑘𝑦 and initial epicycle phase) achievable under the Julian-Toomre (JT) equation (equation 41) a function of 𝑄dr for a many-component stellar disc with constant SFR and AVR 𝜎(𝜏) ∝ 𝜏 𝛽 , for various values of 𝛽. Note that …
Figure 10
Figure 10. Figure 10: ΔGLB as a function of 𝑄0 for 𝑄𝛿 with various values of 𝛿, for two-component discs of stars and gas with various mass ratios and sound-speed/velocity dispersion ratios between the two components. The ΔGLB values seem to be most consistent and overall closest to zero fo…
Figure 11
Figure 11. Figure 11: Contours of constant 𝑄𝛿=5/3 for two-component discs of stars and gas with various mass ratios between the components, as a function of the 1 single-component 𝑄 values of the gas and stellar components, 𝑄g and 𝑄s. The axisymmetric stability criterion is preserved as 𝑄𝛿…
Figure 12
Figure 12. Figure 12: Error in 𝑄𝛿=5/3,RW as an approximation of 𝑄𝛿=5/3 , for 2-component discs of stars and gas with various mass ratios between the two components, as a function of the single-component 𝑄 values of the gas and stellar components, 𝑄g and 𝑄s. Note: at 𝑄g ≈ 0 the error goes m…
Figure 13
Figure 13. Figure 13: LSK dispersion relation (Rafikov 2001) of the Galactic disc at the Solar Neighbourhood, using the gas properties in [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Maximum amplification under the GLB equation, given the properties of the Solar Neighbourhood stated in Section 5.1 (including [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: Maximum swing amplification factor obtainable under the GLB equation for a disc consisting only of the gas components of the Solar Neigh￾bourhood with the properties listed in [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: JWST/MIRI image of M74 [image credit: ESA/Webb, NASA & CSA, J. Lee and the PHANGS-JWST Team. Acknowledgement: J. Schmidt]. We have overlaid a circle with 12 notches at a radius close to 2.6 kpc (the region for which we have listed properties in [PITH_FULL_IMAGE:figur…
Figure 17
Figure 17. Figure 17: Maximum swing amplification factor obtainable under the GLB equation, as a function of 𝑚, given the properties of M74 at 2.6 kpc from the galactic centre listed in Section 5.2 (include [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Galactokinetics II: Spiral structure

    astro-ph.GA 2025-07 conditional novelty 7.0 of 10

    A single linear response kernel for galactic disks reproduces Lindblad-Kalnajs waves, swing amplification, groove instabilities, and Lin-Shu-Kalnajs modes as limiting cases, with smooth connections between them.

  2. Swing amplification in star-gas disks

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    The first explicit linear theory of swing amplification in coupled star–gas disks shows transient wave growth of 10–100x in formally stable disks, governed by distance to the axisymmetric stability boundary.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.