Vertical Structure and Dynamics of a Galactic Disk

Chanda J. Jog

REVIEW 3 major objections 4 minor 1 cited by

Vertical Structure and Dynamics of a Galactic Disk

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A galactic disk's vertical structure is set by the coupled gravity of stars, gas, and dark halo, not by a single stellar component alone.

desk verdict A useful pedagogical review of the author's own multi-component disk plus halo model, but the abstract overstates the flaring claim and the review is narrower than 'comprehensive'. read the letter →

arxiv 2507.02062 v1 pith:L76ZSHGK submitted 2025-07-02 astro-ph.GA

Chanda J. Jog This is my paper

classification astro-ph.GA

keywords galaxies:structurekinematicsanddynamicsISMhalosgalaxy:diskverticalstellarflaring

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 review argues that the vertical structure of a galactic disk is set by the joint, self-consistent gravity of a multi-component disk (stars, HI, and H2 gas) embedded in a dark matter halo, rather than by the classic single-component isothermal disk. Solving the coupled vertical hydrostatic and Poisson equations for all components at once yields stellar density profiles that are more concentrated toward the mid-plane, steeper than the standard $\mathrm{sech}^2$ profile, and consistent with modern observations. The same model explains why the inner HI layer is nearly constant in thickness and why the stellar disk flares by a factor of a few within the optical disk. It also turns observed HI scale heights into constraints on the dark matter halo's density profile and shape.

What carries the argument

The central object is the coupled vertical hydrostatic-Poisson system for a gravitationally coupled multi-component disk plus halo: for each isothermal component the vertical pressure gradient balances the summed vertical gravitational forces of all components and the halo, while the disk Poisson equation sums the vertical second derivatives of the component potentials. The key quantity is the redistributed vertical density profile $\rho_i(z)$ of each component, whose half-width at half-maximum defines the disk scale height. The operative identity is that hydrostatic support of a component depends only on that component's own velocity dispersion, whereas the gravitational force it feels is the sum over all components; hence a thin gas layer or a massive halo compresses any higher-dispersion component embedded in it.

What would settle it

Measure the vertical random motions of old disk stars at several radii between 17 and 25 kpc in the Milky Way with modern astrometric and spectroscopic data; if these motions continue to decline below the ~7 km/s gas value instead of flattening, the sharp flare predicted by the model beyond 17 kpc is not supported. A second decisive test is deep edge-on imaging of a flaring-candidate galaxy fit without the constant-scale-height assumption, to see whether a single $\mathrm{sech}^2$ profile with constant thickness actually fits the full radial range.

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Extended reading notes

Core claim

The central claim is that each disk component's vertical density profile is not a free functional fit but the solution of the coupled joint Poisson-hydrostatic balance equations for stars, HI, and H2 in the halo's gravitational field. The additional vertical force from the thin, low-dispersion gas layers and from the dark matter halo pulls the stellar distribution back toward the mid-plane, so the coupled stellar profile has a higher mid-plane density, a smaller scale height, and a steeper fall-off than the stars-alone $\mathrm{sech}^2$ case. Gas provides the dominant pinching in the inner disk and the halo in the outer disk. The model's outer-disk solution predicts a stellar scale height that grows by roughly a factor of three between 4 and 22 kpc, and, when matched to observed HI scale heights and rotation curves, a dark matter halo with a steeply falling density profile rather than a shallow isothermal cusp.

Load-bearing premise

The sharp outer-disk flare beyond about 17 kpc depends on the assumption that the random vertical motions of stars stop declining at the gas-dispersion floor and stay constant; if those motions keep falling with radius, the predicted flaring curve would be substantially different.

Editorial extensions

If this is right

The routinely assumed $\mathrm{sech}^2$ vertical profile with a constant scale height is not a reliable description of a real galactic disk; the coupled model gives steeper profiles whose scale height increases with radius.
Including gas self-gravity and a small radial gradient in the HI velocity dispersion reproduces the nearly constant HI scale height observed in the inner Milky Way, resolving an old puzzle.
Beyond about 17 kpc the dark matter halo's vertical force dominates, compressing both stars and HI gas; this lowers the HI thickness by factors of 3-4 and makes the outer disk more resistant to tidal distortion.
Using the observed HI scale height together with the rotation curve as joint constraints selects a dark matter halo with a steeply falling density profile ($p \approx 2$) rather than a standard isothermal halo for the Milky Way.
The same pressure-equilibrium argument predicts a generic stellar-disk flare by a factor of a few within the optical radius whenever the stellar velocity dispersion declines more slowly than the surface density ($R_v > 2 R_D$).

Reading between the lines

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

Inference: Existing single-component fits to edge-on galaxies that assume $\mathrm{sech}^2$ and constant scale height likely underestimate how much the stellar layer thickens outward, so derived disk masses and mass-to-light ratios from such fits may need revisiting.
Inference: The sharp outer-disk flare predicted beyond 17 kpc rests on the assumption that stellar random motions stop declining at the gas-dispersion floor; a direct measurement of those motions at larger radii is the cleanest way to test that hinge.
Inference: With integral-field stellar kinematics, the same model could be inverted for external galaxies to predict radial scale-height maps, offering a stacking-free way to look for flaring in edge-on systems.
Inference: Because the halo's vertical compression raises mid-plane gas density at large radii, the model points to a gravitational mechanism for triggering star formation in outer disks, independent of spiral-arm compression.

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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

Quick take: this is not a research paper; it is a review of the Jog/Narayan/Sarkar line of work on vertical disk structure. That is fine, and the review is useful. It sets up the Spitzer sech^2 solution, then walks through the coupled Poisson-hydrostatic equations for stars + HI + H2 in a halo, and shows how extra vertical gravity from gas (inner) and halo (outer) constrains the stellar distribution, giving steeper profiles and reduced scale heights. The application of HI scale heights to constrain halo shape and density profile is a nice, concrete use. The exposition is clear and the equations are standard.

What is not new: no new derivations or data. The core results come from Narayan & Jog 2002, Sarkar & Jog 2018, and related papers. That is acceptable for a review, as long as it is labeled as such.

Soft spots. First, the abstract's 'factor of few within the visible radial extent' overstates Fig. 12. In that figure the stellar HWHM grows by only about 50% between R = 4 and 16 kpc, while the optical radius is roughly 4RD ~ 13 kpc. The factor of 3.3 is reached only at R = 22 kpc, outside that range, and it is driven by the imposed saturation of the stellar vertical velocity dispersion at the gas value beyond 17 kpc. The paper does flag this 'by choice,' but the abstract's robust tone loses that caveat. Second, some model inputs are tuned to match the data (the HI dispersion gradient, halo profile parameters), so the agreement is partly constrained. The central mechanism—gas and halo add vertical gravity—is not in doubt; it follows from the equations with observed surface densities. Third, 'comprehensive' is a stretch: the review is comprehensive on one family of models and touches on related topics (warps, breathing modes, the Gaia phase-space spiral) only briefly. The citation pattern is naturally dominated by the author's own papers; for a review of that work this is acceptable.

Who this is for: a graduate student or someone entering vertical-disk dynamics who wants a single place to see the multi-component formalism and its results. I would go to the original papers for details, but this review is a decent entry point.

Recommendation: I would let it go to serious peer review, because the pedagogical value is real and the faults are correctable. Ask for an abstract rewrite and an explicit caveat: the sharp outer flaring depends on the dispersion saturation, and the flaring factor inside the optical disk is moderate (~1.5), not 'a factor of few.'

Referee Report

3 major / 4 minor

Summary. This review presents the vertical structure of galactic disks in hydrostatic equilibrium, centering on the multi-component disk plus halo model: stars, HI, and H2 are treated as gravitationally coupled isothermal components embedded in a dark-matter halo. The paper derives the classical sech^2 result, generalizes it to the coupled system, applies the model to the Milky Way, NGC 891, NGC 4565, UGC 7321, and dwarf galaxies, and argues that gas and halo vertically constrain the stellar distribution, that stellar disks flare radially, and that HI scale heights can be used as a constraint on the dark-matter halo profile. The abstract claims that a typical stellar disk flares by a factor of a few within the visible radial extent and that this strongly questions the sech^2 profile and constant scale height.

Significance. If the central mechanism is correct, the paper identifies a genuinely important physical effect: the additional vertical gravity from gas and dark matter compresses the stellar distribution, making it steeper than the standard one-component sech^2 profile and explaining observed near-midplane excesses without an ad hoc density profile. The derivation in Sections 4.2.1-4.2.2 is standard and clearly presented, the numerical procedure is sufficiently detailed to be reproducible, and the review usefully collects observational caveats, e.g., the projection problems in edge-on galaxy studies (Section 2.1.1). The paper is less strong as a quantitative prediction: the headline flaring factor depends on a hand-imposed saturation of the stellar velocity dispersion, and some of the claimed observational agreement arises from fitting the same data that later serve as validation.

major comments (3)

[Abstract; Sections 4.2.3 and 4.2.6; Fig. 12; Table 2] The abstract's claim that a typical stellar disk flares by a factor of a few within the visible radial extent is not supported by the model's own Fig. 12 inside the optical disk. The text of Section 4.2.6 states that the HWHM increases by roughly 50% from R=4 to 16 kpc, while the factor of 3.3 quoted from Table 2 occurs at R=22 kpc, well beyond the nominal optical radius of about 4 R_D ~ 13 kpc for R_D=3.2 kpc. The steep flaring beyond R=17 kpc is produced directly by the imposed saturation of the stellar vertical velocity dispersion at the gas-dispersion value (Section 4.2.3), not by the coupled Poisson-hydrostatic equations themselves. If the true stellar dispersion continues to decline beyond 17 kpc rather than saturating, or saturates at a different radius, the flaring factor changes substantially. The physical constraining mechanism is not in question; the quantitative 'factor of few within the visible disk' claim should be revised or explicitly conditioned on the adopted saturation assumption.
[Sections 4.2.4 and 4.3.2] Some of the claimed agreement between model and observation is partly circular, because input parameters are tuned to reproduce the same data. In Section 4.2.4, the HI vertical velocity dispersion gradient is chosen specifically to give the best fit between the model HWHM and the observed HI scale heights; in Section 4.3.2, the halo density parameters are obtained by requiring the model HI scale heights and rotation curve to match the observed values. The paper does repeatedly use the language of 'constraints' and 'best-fit', which is honest, but when the same Section summarizes the result as 'in good agreement with observations', that agreement is calibration rather than prediction. The robust part of the paper, namely that gas and halo add vertical gravity and steepen the stellar profile, follows from the equations and observed surface densities and does not depend on this circularity; the halo-shape and halo-profile conclusions, however, should be presented as fits with a stated degeneracy, not as independent determinations.
[Section 5.2 and Section 5.3.1] The discussion of 'sharp flaring in the outer disk' presents the saturation of the stellar velocity dispersion as physically motivated because stars cannot have a lower dispersion than the gas from which they form. This argument is plausible but not a measured constraint in the outer disk, and Section 5.2 does not quantify the sensitivity of the flaring factor to the saturation radius or to the value of the floor dispersion. A test would be to rerun the model with the dispersion declining as an exponential without saturation, or with saturation at R=20-25 kpc, and to show how Fig. 12 and Table 2 change. Until such a sensitivity analysis is provided, the sharp outer-disk flaring should not be presented as a generic, robust prediction.

minor comments (4)

[Section 4.2.7, Fig. 16 caption] The text says the scale height increases linearly from 0.75 to 1.25 kpc, while the Fig. 16 caption says from 0.5 to 1.25 kpc; the two values should be harmonized.
[Section 4.3.2, UGC 7321 paragraph] The text states that observations are known up to R=5.6 pc; this should be 5.6 kpc.
[Section 4.2.6, Fig. 13 and surrounding text] The notation 'exponent 2/n' is used for the ordinate of Fig. 13, while the text sometimes refers to the best-fit exponent and sometimes to the parameter n itself; a sentence defining 'exponent 2/n' in terms of Eq. (4.7) would remove ambiguity.
[Section 4.3.1, Eq. (4.12)-(4.13)] For the flat-rotation-curve thick disk case, Eq. (4.13) looks identical to the thin-disk equation except that the halo density appears in the source term; it would help the reader to state explicitly that the disk and halo Poisson equations remain coupled through the radial-term condition even when the net radial term is zero.

Circularity Check

2 steps flagged · score 4.0 of 10

The central gas-plus-halo constraining mechanism follows independently from the coupled Poisson-hydrostatic equations, but the quantitative flaring claim is partly built in through the hand-imposed saturation of the stellar velocity dispersion at R=17 kpc, and the inner-Galaxy HI scale-height agreement is partly a fit to the same data.

fitted input called prediction [Section 4.2.4, 'Results for vertical scale heights in the inner Galaxy' (discussion of Fig. 7)]
"A small linear gradient of -0.8 km s−1 kpc−1 was tried, starting with a value of HI velocity dispersion of 8 km s−1 at R = 8.5 kpc. The resulting scale height values vs. R for HI, H2 and stars are given in the three panels in Fig 7. This value of the velocity gradient was chosen since it was found to give the best-fit between the model HWHM and the observed values for HI for the range 2-12 kpc."

The radial gradient of the HI velocity dispersion is a free input that is tuned to minimize the difference between the model HWHM and the observed HI scale heights over 2-12 kpc. The subsequent statements that the model 'shows an overall agreement with observations' and 'explains the old puzzle of the nearly constant HI scale height' are therefore not independent tests of the model: a different adopted gradient would produce a different degree of agreement. The underlying coupled-equation mechanism is not circular, but this particular quantitative validation reduces to a fit of the same data it claims to reproduce.
self definitional [Section 4.2.6, 'Detailed results for stellar density distribution in the outer Galaxy' (Fig. 12 and Table 2), and Abstract]
"Beyond R = 17 kpc, the stellar disk flares steeply because the stellar velocity dispersion saturates (by choice) to the gas dispersion value beyond this radius (see Section 4.2.3). ... A typical stellar disk is shown to flare by a factor of few within the visible radial extent of the disk."

The sharp flaring beyond 17 kpc is not derived from the coupled Poisson-hydrostatic equations alone; it is imposed by the adopted saturation of the stellar vertical velocity dispersion at R=17 kpc. The abstract's 'factor of few within the visible radial extent' is only reached at R=22 kpc (factor 3.3 in the quoted Table 2), while within the optical disk (4RD~13 kpc) Fig. 12 shows an increase of only about 50%. The headline flaring claim is therefore largely equivalent to the hand-imposed input prescription rather than a standalone prediction of the model.

full rationale

The paper's central mechanism—that gas and dark-matter-halo gravity vertically constrain the stellar distribution, producing a steeper profile than the stars-alone sech^2 case—follows directly from solving the joint Poisson and hydrostatic-balance equations with observed surface densities and dispersions, and is not circular. However, two quantitative claims are partially circular. First, the inner-Galaxy HI scale-height agreement is obtained after tuning the HI velocity dispersion gradient to give the best fit to exactly those observed scale heights (Section 4.2.4), so the agreement is partly a fit. Second, the strong outer-disk flaring and the abstract's 'factor of few within the visible radial extent' depend on the hand-imposed saturation of the stellar velocity dispersion at R=17 kpc (Section 4.2.6), which the paper itself describes as 'by choice'; within the optical radius the model's own figure shows only a modest ~50% increase, so the generic flaring statement overstates what is derived from the equations alone. These are partial circularities in specific quantitative validations, not a collapse of the whole derivation chain; the physical constraining effect and the halo-constraint application have independent content. Hence a moderate score of 4 is appropriate.

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

The model is a boundary-value problem requiring inputs for surface densities and velocity dispersions. Many inputs come from observations, but several key parameters (HI dispersion gradient, Rv, saturation radius, halo profile) are tuned to match the very observations the model is then said to explain. This does not invalidate the central mechanism, but it reduces the strength of the claimed agreement.

free parameters (6)

HI vertical velocity dispersion gradient = -0.8 km/s/kpc
Chosen to give the best-fit of model HI scale heights to observed values (Section 4.2.4).
Stellar velocity dispersion radial scale length Rv (Milky Way) = 8.7 kpc
Taken from observed radial velocity dispersion data [56], but the vertical-to-radial ratio is assumed constant at 0.45.
Saturation radius for stellar velocity dispersion = 17 kpc
Assumed so that stellar dispersion does not fall below gas dispersion; drives outer disk flare (Section 4.2.3).
Halo density profile parameters (p, rho0h, Rc) for outer Galaxy = p=2, rho0h=0.035 Msun/pc^3, Rc=9.4 kpc
Best-fit to observed rotation curve and HI scale heights (Section 4.3.2).
Rv/RD for NGC 891 and NGC 4565 = 2-2.5 and 2.5-3 respectively
Constrained from the observed spread in scale height (Section 4.2.7).
Rv for UGC 7321 = 3.2 RD
Fitted to match observed stellar scale height vs. R (Section 4.3.2).

assumptions (6)

domain assumption Each disk component (stars, HI, H2) is isothermal and in vertical hydrostatic equilibrium.
Used throughout Section 4.1-4.2; velocity dispersion constant along z for each component.
domain assumption The disk is thin: only the z-term in the Poisson equation matters for the disk.
Section 3.1 and 4.2.1; this justifies the local 1-D treatment. It is relaxed in Section 4.3.
domain assumption The dark matter halo is unaffected by the disk and acts as a fixed external potential.
Section 4.2.1, after Eq. (4.2); the halo is treated as a reservoir.
domain assumption The disk components are co-spatial, concentric, and coplanar.
Section 4.1.2; this simplifies the coupled equations.
domain assumption The stellar velocity ellipsoid is aligned with spherical polar coordinates, so the cross term vRvz = (sigma_R^2 - sigma_z^2)(z/R).
Section 4.4.1, after Eq. (4.16); used in the complete Jeans model.
standard math Standard Newtonian gravity and the Poisson equation apply.
Throughout.

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Cite this review

Pith. "Pith review of Vertical Structure and Dynamics of a Galactic Disk." pith.science (2026). https://pith.science/paper/L76ZSHGK

@misc{pith2026250702062,
  author       = {Pith},
  title        = {Pith review of: Vertical Structure and Dynamics of a Galactic Disk},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L76ZSHGK}},
  note         = {Machine review of arXiv:2507.02062}
}

read the original abstract

Most of the visible mass in a typical spiral galaxy is distributed in a thin disk, with a radial extent much larger than its thickness. While the planar disk structure, including non-axisymmetric features such as spiral structure, has been studied extensively, the vertical structure has not received comparable attention. This review aims to give a comprehensive, pedagogic introduction to the rich topic of vertical structure of a galactic disk in hydrostatic equilibrium and discuss the theoretical developments in this field in the context of recent observations. A realistic multi-component disk plus halo model of a galaxy has been developed and studied by us in detail. This takes account of both stars and interstellar gas, treated as isothermal components with different velocity dispersions, which are gravitationally coupled; further, the disk is in the gravitational field of the dark matter halo. This review focuses on this model and the results from it in different physical cases. The gas and halo crucially affect the resulting self-consistent stellar distribution such that it is vertically constrained to be closer to the mid-plane and has a steeper profile than in the standard one-component case, in agreement with modern observations. A typical stellar disk is shown to flare by a factor of few within the visible radial extent of the disk. These robust results question the sech^2 profile and a constant scale height, routinely used in the literature, for convenience. In an important application, the observed HI gas scale height is used as a constraint on the model which helps determine the shape and the density profile of the dark matter halo for galaxies. Finally, we outline some key, open questions which can be addressed in the near future using the above model, and new observational data -- for example, from IFU surveys and JWST -- for a better understanding of this topic.

Figures

Figures reproduced from arXiv: 2507.02062 by the authors.

**Figure 2.** Plot of |Kz|, the vertical gravitational force per unit mass due to a cloud complex, and that due to the undisturbed stellar disc, versus z, the distance from the mid-plane, at the complex centre. For z < 200 pc, the force due to the complex dominates over that due to the disk. The ratio is a maximum, equal to 9.5, at the outer edge of the complex, at z = 60 pc. Source: Taken from [112] ∂(Kz)s ∂z + ∂(Kz)complex ∂z +… view at source ↗

**Figure 3.** Plot of self-consistent density for self-gravitating, undisturbed stellar disk [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗

Figures from the paper (32 more)

**Figure 4.** Figure 4: Plot of vertical scale height (HWHM) for the modified stellar distribution for [PITH_FULL_IMAGE:figures/full_fig_p032_4.png]

**Figure 5.** Figure 5: Schematic diagram of the distribution of the three disk components: namely, [PITH_FULL_IMAGE:figures/full_fig_p037_5.png]

**Figure 6.** Figure 6: The plot of scale height (HWHM) vs. R for interstellar HI, H [PITH_FULL_IMAGE:figures/full_fig_p045_6.png]

**Figure 7.** Figure 7: The plot of scale height (HWHM) vs. R for HI, H [PITH_FULL_IMAGE:figures/full_fig_p047_7.png]

**Figure 8.** Figure 8: Plot of vertical self-gravitational force per unit mass, [PITH_FULL_IMAGE:figures/full_fig_p050_8.png]

**Figure 9.** Figure 9: Self-consistent vertical density for stars vs. [PITH_FULL_IMAGE:figures/full_fig_p051_9.png]

**Figure 10.** Figure 10: Plot of gravitational force per unit mass, [PITH_FULL_IMAGE:figures/full_fig_p052_10.png]

**Figure 11.** Figure 11: Self-consistent vertical density of stars vs. [PITH_FULL_IMAGE:figures/full_fig_p053_11.png]

**Figure 12.** Figure 12: Plot of the model scale height (HWHM) of the vertical stellar density distribu [PITH_FULL_IMAGE:figures/full_fig_p055_12.png]

**Figure 13.** Figure 13: Plot of the best-fit exponent 2/n vs. |∆z| for the range of z values over which the model vertical density distribution for stars is fitted by a distribution of type sech2/n; shown for R = 6 kpc. The value of n varies with |∆z|, hence n is not a robust indicator of th…

**Figure 14.** Figure 14: Plot of the best-fit value of the exponent 2 [PITH_FULL_IMAGE:figures/full_fig_p058_14.png]

**Figure 15.** Figure 15: Self-consistent vertical density distribution of HI gas vs. [PITH_FULL_IMAGE:figures/full_fig_p060_15.png]

**Figure 16.** Figure 16: Plot of surface brightness, I vs. z for NGC 891 from [22] who used their data and obtained this composite z-profile (the top curve in this figure) by vertically shifting the individual z-profiles at different R into coincidence at z’= 1.5 kpc. The vertical bars indica…

**Figure 17.** Figure 17: Plot of resulting stellar scale height (HWHM) vs. [PITH_FULL_IMAGE:figures/full_fig_p065_17.png]

**Figure 18.** Figure 18: Plot of calculated HI scale height (HWHM) vs. R in the outer Galaxy. The [PITH_FULL_IMAGE:figures/full_fig_p075_18.png]

**Figure 19.** Figure 19: Plot of HI scale height vs.R in the outer Galaxy. The best fit to the data [PITH_FULL_IMAGE:figures/full_fig_p077_19.png]

**Figure 20.** Figure 20: Plot of results for the HI scale height (HWHM) vs. R for two dwarf galaxies [PITH_FULL_IMAGE:figures/full_fig_p079_20.png]

**Figure 21.** Figure 21: Plot of resulting stellar vertical density, [PITH_FULL_IMAGE:figures/full_fig_p087_21.png]

**Figure 22.** Figure 22: Plot of resulting self-consistent vertical density distribution, [PITH_FULL_IMAGE:figures/full_fig_p092_22.png]

**Figure 23.** Figure 23: The non-isothermal density vertical density distribution of stars versus [PITH_FULL_IMAGE:figures/full_fig_p094_23.png]

**Figure 24.** Figure 24: The resulting non-isothermal vertical stellar distribution obtained for a velocity [PITH_FULL_IMAGE:figures/full_fig_p095_24.png]

**Figure 25.** Figure 25: Plot of the vertical force per unit mass, [PITH_FULL_IMAGE:figures/full_fig_p106_25.png]

**Figure 26.** Figure 26: Plot of the integrand −zρ(z)Kz of the gravitational potential energy per unit area versus z for stars and gas (HI) at R= 8.5 kpc; obtained for a one-component case and then for the coupled case. The left panel shows the plot of energy integrand for stars-alone (solid …

**Figure 27.** Figure 27: Plot of work done, Ez, to raise a unit test mass from the mid-plane to a vertical height z versus z, at R=8.5 kpc: for the stars-alone case against its self-gravity (solid curve); for gas-alone case against its own self-gravity (dashed line); and for stars or gas in t…

**Figure 28.** Figure 28: Calculated vertical scale height for the atomic hydrogen gas (HI) (solid line) [PITH_FULL_IMAGE:figures/full_fig_p129_28.png]

**Figure 29.** Figure 29: Resulting prolate-shaped isodensity contours (on the R-z plane) of the best-fit [PITH_FULL_IMAGE:figures/full_fig_p130_29.png]

**Figure 30.** Figure 30: A schematic diagram of bending waves and breathing modes, which illustrates [PITH_FULL_IMAGE:figures/full_fig_p146_30.png]

**Figure 31.** Figure 31: A striking example of a stellar warp in the galaxy ESO 510-G13. The image [PITH_FULL_IMAGE:figures/full_fig_p148_31.png]

**Figure 32.** Figure 32: Dust corrugations in NGC 4302: The top left panel shows the Spitzer/IRAC [PITH_FULL_IMAGE:figures/full_fig_p149_32.png]

**Figure 33.** Figure 33: Breathing motions: Plot of mean velocity [PITH_FULL_IMAGE:figures/full_fig_p151_33.png]

**Figure 34.** Figure 34: Phase space spiral in the Milky Way [343]: Plot of distribution of stars in the vertical position-velocity (z − vz) phase space for stars sampled within the galactocentric radial range of 8.24-8.44 kpc; coloured as a function of median azimuthal velocity, vϕ, in bins …

**Figure 35.** Figure 35: CALIFA Survey results for stellar velocity dispersion in galaxies [ [PITH_FULL_IMAGE:figures/full_fig_p157_35.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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