Pith. sign in

REVIEW 3 major objections 8 minor 147 references

In short-wavelength local limit, stratified hydrostatic media still obey the classical Jeans criterion, and all local perturbations are stable.

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 →

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

T0 review · grok-4.5

2026-07-30 16:30 UTC pith:5UD3AHVS

load-bearing objection Solid local recovery of classical Jeans without the swindle; the universal “all local modes are stable” claim overreaches via a virial step the paper itself undercuts. the 3 major comments →

arxiv 2607.23656 v1 pith:5UD3AHVS submitted 2026-07-26 astro-ph.GA

On the Jeans criterion in hydrostatic stratified media

Mahmood Roshan , Asiyeh Habibi This is my paper
classification astro-ph.GA
keywords Jeans instabilityhydrostatic equilibriumstratified mediadispersion relationgravitational collapselocal perturbationsJeans swindle
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.

The classical Jeans analysis assumes a uniform background, an idealization that is inconsistent with gravity (the so-called Jeans swindle). This paper asks what changes when the background is a realistic hydrostatic stratified fluid with nonzero density and pressure gradients. Using linear Eulerian perturbations plus a spatial averaging procedure around a local patch, the authors derive a modified dispersion relation in which those gradients enter in a complicated, direction-dependent way. No single stability bound therefore exists for an arbitrary hydrostatic background. In the short-wavelength local limit, however, the gradient corrections become negligible and the ordinary Jeans relation is recovered. Combined with the virial estimate that the system scale itself is comparable to the Jeans length, this implies that every truly local perturbation is automatically Jeans-stable. The result both justifies everyday use of the classical criterion for small-scale modes and explains why a universal local criterion cannot be written down.

Core claim

Background gradients preclude any general local gravitational stability criterion for arbitrary hydrostatic equilibria, yet in the short-wavelength limit kl ≫ 1 the dispersion relation reduces to the classical Jeans form ω² ≃ c_s² k² − 4π G ρ₀, so the standard Jeans criterion remains valid and all local perturbations are stable against local collapse.

What carries the argument

The averaged local dispersion relation (equation 35) obtained by Taylor-expanding background quantities inside a small volume surrounding the perturbation and retaining the resulting α and β gradient corrections; in the kl ≫ 1 limit those corrections drop out and the classical Jeans relation is recovered.

Load-bearing premise

The claim that every local mode automatically satisfies k > k_J because virial equilibrium forces the system scale to be comparable to the Jeans length; this is only an order-of-magnitude argument and the paper itself notes exceptions when the Jeans length is much smaller than the system.

What would settle it

Compute or simulate the growth rate of a short-wavelength density perturbation (kl ≫ 1) placed inside a known hydrostatic stratified equilibrium (e.g., a polytropic sphere or stratified slab) and check whether the measured ω² matches c_s² k² − 4π G ρ₀ rather than the full gradient-corrected expression.

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

If this is right

  • Everyday Jeans analyses of small-scale clumps inside molecular clouds or stellar interiors remain valid even though the background is stratified.
  • No universal local stability formula can replace the classical Jeans criterion for arbitrary hydrostatic backgrounds.
  • Long-wavelength (global) modes lie outside the present analysis and must still be treated case by case.
  • When first-order 1/kl corrections are kept, only wave-vectors perpendicular to the density gradient recover purely real Jeans frequencies; other directions produce overstability or standing waves.

Where Pith is reading between the lines

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

  • The same averaging-plus-short-wave reduction could be repeated for rotating or magnetized hydrostatic backgrounds to test whether classical Toomre or magnetic Jeans criteria likewise survive locally.
  • If high-resolution simulations of stratified clouds systematically find collapse only on scales approaching the system size, that would corroborate the claim that all truly local modes are stable.
  • The direction dependence retained at order 1/kl suggests that anisotropic collapse statistics relative to the local density gradient could be observable in filamentary clouds.

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

3 major / 8 minor

Summary. The manuscript revisits the Jeans stability analysis for a non-rotating barotropic fluid in hydrostatic equilibrium, keeping the background density, pressure, and sound-speed gradients that the classical treatment discards. Linearizing Euler/continuity/Poisson about a hydrostatic background yields a modified wave equation for the density perturbation (Eq. 8) whose Fourier form is a convolution (Eq. 17), from which no background-independent stability criterion can be extracted — correctly, in my view. The authors then introduce a spatial-averaging procedure over a small ball of radius a around the perturbation, with background fields Taylor-expanded to first order, and derive an explicit k-dependent dispersion relation (Eq. 35) involving direction-dependent gradient couplings α_i ~ 1/kl and β_i ~ 1/k²l². In the short-wavelength limit kl ≫ 1 this reduces to the standard Jeans form ω² ≃ c_s²k² − 4πGρ₀, obtained without the Jeans swindle. The paper further claims (i) that the O(1/kl) corrections leave the Jeans criterion intact (Eqs. 40–44), and (ii) that all local perturbations in any hydrostatic stratified system are stable against local gravitational collapse, because locality requires k > 1/L and the virial theorem gives 1/L ~ k_J.

Significance. If the leading-order result holds, it is a useful and honest contribution: a self-contained, parameter-free derivation showing why the classical Jeans criterion survives stratification at leading order, with no fitted constants and a clear identification of why no universal criterion exists for arbitrary backgrounds (the convolution structure of Eq. 17). The explicit dispersion relation (35), with its dependence on wavevector direction and on the gradient scale l, is a concrete result that could be tested against specific equilibria (e.g., Bonnor–Ebert spheres). However, the headline claim in the abstract — that all local perturbations in a hydrostatic system are stable against local collapse — does not follow from the analysis as written and is in tension with a caveat the paper itself states; if left unqualified it would mislead readers, since local Jeans-unstable cores inside globally stable hydrostatic envelopes are precisely the standard picture of star formation the paper cites (Larson 2003; McKee & Ostriker 2007). The leading-order recovery of the Jeans criterion is the solid result; the universal stability claim is not established.

major comments (3)
  1. [§3.2, final paragraph (virial argument for 'all local perturbations are stable')] The inference 'local ⇒ k > 1/L, and 1/L ~ k_J by the virial theorem, therefore k > k_J' mixes two different Jeans wavenumbers. The dispersion relation recovered in this section is local, with k_J(r₀) = √(4πGρ₀(r₀))/c_s evaluated at the perturbation site, whereas the virial estimate 1/L ~ k_J (Binney & Tremaine 2008; Nipoti 2023) involves the global mean density and system size. In any centrally concentrated hydrostatic equilibrium the local k_J exceeds the global 1/L: for a critical Bonnor–Ebert sphere, k_J(center)·R = ξ₁ ≈ 6.45, so there is a broad band 1/L < k < k_J(r₀) of modes that are local by the paper's own definition yet Jeans-unstable — and this is exactly where collapse matters. Additionally, the virial link presumes the background is supported against its own self-gravity; the paper's stated scope is an arbitrary hydrostatic background, which includes externally supported conf
  2. [§3.2, Eqs. (28), (41)–(42): the approximation I ≃ 1 at ka ≃ 1] Eq. (28) gives I(ka) = 3 + k²a²/(ka cot ka − 1). Direct evaluation at ka = 1 (cot 1 ≈ 0.642) gives I(1) ≈ 0.21, not of order unity (the small-argument expansion is I ≈ (ka)²/5). The step from Eq. (41) to Eq. (42) adopts I ≃ 1, which sets (I−1) = 0 and thereby eliminates the gravity-gradient term −4πGρ₀(I−1)α₁ from the O(1/kl) correction. With the actual value I(1) ≈ 0.21, one has (I−2) ≈ −1.79 and (I−1) ≈ −0.79, so the gravity-gradient term survives at the same order as the sound-speed term retained in Eq. (42); near k ~ k_J the two contributions are comparable. Consequently Eq. (42), the claimed identification with sound waves in a stratified medium (Clarke & Carswell 2007), and the mode analysis in Eqs. (43)–(44) are quantitatively incorrect as derived, even though the qualitative conclusion (Jeans criterion intact at leading order, O(1/kl) corrections overstability-like) may survive.
  3. [§3.1, step from Eq. (34) to Eq. (35)] Equation (34) states that a single k-integral of ρ̂₁(ω,k) I₁(ka) e^{ik·r₀} (ω² − B(k)) vanishes; the dispersion relation (35) is then obtained by 'setting the integrand to zero.' Vanishing of the integral does not imply vanishing of the integrand, and ρ̂₁ is not fully arbitrary (the perturbation must be compact inside δV, with ρ₁ = 0 on ∂δV as used for the divergence-theorem step). This step is the load-bearing approximation of the averaging method and deserves an explicit justification — e.g., that the perturbation is a narrow wavepacket in k-space centered on k with width Δk ≪ k, so that the slowly varying factor ω² − B(k) can be evaluated at the packet center. As written, the logical status of the paper's main equation is unclear. Relatedly, I₁(ka) has zeros at tan(ka) = ka (first at ka ≈ 4.49), where I = k²I₂/I₁ diverges and Eq. (35) becomes singular; this is presumably an artifact o
minor comments (8)
  1. [§3.1, after Eq. (25)] The phrase 'we can simplify equation (24)' appears twice in succession (before and after Eq. 26); one instance should be removed.
  2. [§3.2] Typo: 'occure' should be 'occur'.
  3. [After Eq. (16)] Sentence begins with a capitalized 'Applying' mid-sentence following the displayed equation; punctuation/capitalization should be fixed.
  4. [§3.1, divergence-theorem step] The assumption that ρ₁ vanishes on ∂δV is in mild tension with the subsequent treatment of the perturbation as a Fourier mode with ka ≃ 1: a compactly supported perturbation has a broad Fourier transform, which bears on the localization ansatz behind Eq. (34). A brief remark clarifying the hierarchy between the perturbation size, the averaging radius a, and the wavelength would help.
  5. [§3, Eq. (19)] The separable form ρ₁(r,t) = f(r)g(t) used to simplify Eq. (17) is restrictive (it excludes the wavepacket modes used later in §3.1); the relation between the separable-mode analysis and the averaging analysis should be stated.
  6. [§3.1–3.2] A figure showing I(ka) over the relevant range would make the discussion of the I ≃ 1 approximation and the α_i, β_i scalings considerably clearer; the manuscript currently has no figures.
  7. [§3.2, Eq. (36)] ψ_i is defined as the triple (ρ₀, c_s², ∇²c_s²) but β₃ is never used; either drop the third component from the β definition or state explicitly that only β₁, β₂ enter.
  8. [General] A worked example (e.g., the Bonnor–Ebert sphere, for which k_J(r₀) varies by a factor ξ₁ across the system) would concretely illustrate both the validity of the short-wavelength reduction and the limits of the global virial estimate, and would strengthen the paper's contact with the star-formation applications cited in the Introduction.

Circularity Check

0 steps flagged

No significant circularity: dispersion relation and short-wavelength reduction are derived from the fluid equations plus hydrostatic equilibrium without fitted inputs or load-bearing self-citation.

full rationale

The paper’s chain is self-contained. Section 2 argues physically that background gradients are negligible for small perturbations and recovers the classical Jeans equation without the swindle. Section 3 starts from the linearized Euler, continuity, and Poisson equations on a hydrostatic background, obtains the integro-differential equation (8), then an exact Fourier form (17)–(18) and, via a local spatial average plus Taylor expansion of background quantities, the approximate dispersion relation (35). The short-wavelength reduction kl ≫ 1 drops the α_i ∼ 1/kl and β_i ∼ 1/k²l² terms by explicit scaling and recovers ω² ≃ c_s² k² − 4π G ρ₀. No parameter is fitted to data and then re-presented as a prediction; no uniqueness theorem or ansatz is imported from the authors’ prior work; citations (Jeans 1902, Binney & Tremaine, Nipoti 2023/2026) supply background or concurrent comparison, not premises that force the algebra. The further claim that ‘all local perturbations are stable’ rests on the external virial estimate 1/L ∼ k_J; that step may be physically contestable, but it is not circular under the definitions used here. Steps list is therefore empty.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

Load-bearing content is classical self-gravitating barotropic hydrodynamics plus an ad hoc spatial averaging/Taylor scheme. No parameters are fitted to data. No new physical entities are postulated. The result stands or falls on standard continuum assumptions and on the validity of the local averaging approximations.

axioms (7)
  • domain assumption Background is a non-rotating barotropic fluid in exact hydrostatic equilibrium: ∇p0/ρ0 = −∇Φ0, with p = p(ρ).
    Stated at the opening of §2 and used throughout to replace pressure gradients by gravity and to close the sound-speed terms.
  • domain assumption Linear Eulerian perturbations are sufficient; products of first-order quantities are dropped.
    Standard linearization leading to equations (4)–(6) and (8).
  • domain assumption For perturbations small compared with the host system, |∇p0| ≪ |∇p1| and |∇Φ0| ≪ |∇Φ1|, so background terms may be dropped in the momentum equation without a Jeans swindle.
    Physical argument of §2; underpins the claim that the classical criterion needs no swindle for small-scale modes.
  • ad hoc to paper Spatial averages over a small ball δV of radius a around the perturbation, with background fields Taylor-expanded to first order in x and with ρ1 vanishing on ∂(δV), faithfully capture local Eulerian dynamics.
    Core of §3.1; the divergence-theorem vanishing of the fourth term in (8) and the appearance of I1, I2, α_i, β_i all rest on this construction.
  • ad hoc to paper In the short-wavelength regime kl ≫ 1 one may set I(ka) ≃ 1 and drop O(1/k²l²) terms, recovering ω² ≃ c_s²k² − 4πGρ0.
    §3.2 approximations that convert the complicated relation (35) into the standard Jeans criterion.
  • domain assumption Virial equilibrium of the host implies 1/L ∼ k_J, hence local modes with k > 1/L automatically satisfy k > k_J.
    Invoked in §3.2 citing Binney & Tremaine 2008 and Nipoti 2023 to conclude all local perturbations are stable.
  • standard math Poisson gravity in Newtonian form ∇²Φ = 4πGρ and Fourier representation Φ̂ = −4πG ρ̂/k² are valid for the systems considered.
    Used from equation (3) onward; standard in the subfield.

pith-pipeline@v1.2.0-grok45-kimik3 · 14298 in / 3762 out tokens · 90696 ms · 2026-07-30T16:30:08.734748+00:00 · methodology

0 comments
read the original abstract

The classical Jeans stability criterion neglects spatial gradients in background physical quantities such as density and pressure. Here, we revisit the Jeans analysis for a non-rotating fluid in hydrostatic equilibrium, explicitly accounting for these gradients and deriving a modified dispersion relation governing perturbation propagation. Using physical arguments, we show why the Jeans swindle is not required to derive the Jeans criterion for perturbations that are small compared to the size of the host system. We use linear perturbation analysis alongside an averaging procedure to study the behavior of the local Eulerian perturbations in a hydrostatic medium, showing that background gradients enter the dispersion relation in a highly non-trivial manner, precluding the derivation of a general gravitational stability criterion applicable to an arbitrary hydrostatic system. We demonstrate that, in the local short-wavelength limit, the standard Jeans criterion remains valid despite the presence of nonzero background gradients, and that all local perturbations in a hydrostatic system are stable against local gravitational collapse. We conclude that it is not possible to derive a general local gravitational stability criterion valid for an arbitrary hydrostatic background; however, in the local short-wavelength limit, the standard Jeans criterion remains valid.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

147 extracted references · 16 canonical work pages · 2 internal anchors

  1. [1]

    Philosophical Transactions of the Royal Society of London Series A , year = 1902, month = jan, volume =

    The Stability of a Spherical Nebula. Philosophical Transactions of the Royal Society of London Series A , year = 1902, month = jan, volume =. doi:10.1098/rsta.1902.0012 , adsurl =

    arXiv 1902

  2. [2]

    Galactic Dynamics: Second Edition

  3. [3]

    Jeans swindle

    The “Jeans swindle”: A true story—mathematically speaking , journal =. 2003 , issn =. doi:https://doi.org/10.1016/S0196-8858(02)00556-0 , url =

    work page doi:10.1016/s0196-8858(02)00556-0 2003

  4. [4]

    , keywords =

    Why does the jeans swindle work?. , keywords =. doi:10.1093/mnrasl/sls051 , archivePrefix =. 1210.3363 , primaryClass =

    Pith/arXiv arXiv

  5. [5]

    , keywords =

    Varying-G gravity: physical properties, asymptotic regimes, and Green's functions, an event horizon, the vacuum energy density, and the external pressure that modifies Jeans stability. , keywords =. doi:10.1093/mnras/stac3416 , adsurl =

    work page doi:10.1093/mnras/stac3416

  6. [6]

    10.1051/0004-6361/202557742

    The Jeans criterion for hydrostatic and infalling gas , DOI= "10.1051/0004-6361/202557742", url= "https://doi.org/10.1051/0004-6361/202557742", journal =

    work page doi:10.1051/0004-6361/202557742

  7. [7]

    , keywords =

    Local gravitational instability of stratified rotating fluids: three-dimensional criteria for gaseous discs. , keywords =. doi:10.1093/mnras/stac3403 , archivePrefix =. 2211.09831 , primaryClass =

    Pith/arXiv arXiv

  8. [8]

    Reports on Progress in Physics , keywords =

    The physics of star formation. Reports on Progress in Physics , keywords =. doi:10.1088/0034-4885/66/10/R03 , archivePrefix =. astro-ph/0306595 , primaryClass =

    Pith/arXiv arXiv

  9. [9]

    , keywords =

    Theory of Star Formation. , keywords =. doi:10.1146/annurev.astro.45.051806.110602 , archivePrefix =. 0707.3514 , primaryClass =

    Pith/arXiv arXiv

  10. [10]

    , year = 1957, month = jan, volume =

    Jeans' formula for gravitational instability. , year = 1957, month = jan, volume =. doi:10.1093/mnras/117.1.104 , adsurl =

    work page doi:10.1093/mnras/117.1.104 1957

  11. [11]

    doi:10.1007/978-3-642-30304-3 , adsurl =

    Stellar Structure and Evolution. doi:10.1007/978-3-642-30304-3 , adsurl =

    doi:10.1007/978-3-642-30304-3

  12. [12]

    , year = 1964, month = aug, volume =

    The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity. , year = 1964, month = aug, volume =. doi:10.1086/147938 , adsurl =

    doi:10.1086/147938 1964

  13. [13]

    Principles of Astrophysical Fluid Dynamics

  14. [14]

    , volume = 129, pages = 243, doi =

  15. [15]

    A&A , publisher =

    A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical magnetohydrodynamics , author =. A&A , publisher =. doi:10.1051/0004-6361:20065371 , url =

    doi:10.1051/0004-6361:20065371

  16. [16]

    A&A , publisher =

    Cosmological hydrodynamics with adaptive mesh refinement , author =. A&A , publisher =. doi:10.1051/0004-6361:20011817 , url =

    doi:10.1051/0004-6361:20011817

  17. [17]

    An implicit scheme for solving the anisotropic diffusion of heat and cosmic rays in the

    Yohan Dubois and Beno. An implicit scheme for solving the anisotropic diffusion of heat and cosmic rays in the. A&A , publisher =. doi:10.1051/0004-6361/201527126 , url =

    doi:10.1051/0004-6361/201527126

  18. [18]

    doi:10.1093/mnras/stt1698 , keywords =

    , volume = 436, number = 2, pages =. doi:10.1093/mnras/stt1698 , keywords =

    doi:10.1093/mnras/stt1698

  19. [19]

    doi:10.1051/0004-6361:20031117 , keywords =

    , volume = 409, pages =. doi:10.1051/0004-6361:20031117 , keywords =

    doi:10.1051/0004-6361:20031117

  20. [20]

    , volume = 285, pages =

  21. [21]

    doi:10.1103/PhysRev.75.1169 , adsurl =

    Physical Review , volume = 75, number = 8, pages =. doi:10.1103/PhysRev.75.1169 , adsurl =

    doi:10.1103/physrev.75.1169

  22. [22]

    doi:10.1093/mnras/182.2.147 , keywords =

    , volume = 182, pages =. doi:10.1093/mnras/182.2.147 , keywords =

    doi:10.1093/mnras/182.2.147

  23. [23]

    doi:10.1146/annurev.aa.22.090184.002233 , adsurl =

    , volume = 22, pages =. doi:10.1146/annurev.aa.22.090184.002233 , adsurl =

    arXiv

  24. [24]

    doi:10.1016/0370-1573(87)90134-7 , adsurl =

    , volume = 154, number = 1, pages =. doi:10.1016/0370-1573(87)90134-7 , adsurl =

    doi:10.1016/0370-1573(87)90134-7

  25. [25]

    doi:10.1146/annurev.aa.15.090177.001135 , keywords =

    , volume = 15, pages =. doi:10.1146/annurev.aa.15.090177.001135 , keywords =

    arXiv

  26. [26]

    doi:10.1093/mnras/stt179 , keywords =

    , volume = 431, number = 1, pages =. doi:10.1093/mnras/stt179 , keywords =

    doi:10.1093/mnras/stt179

  27. [27]

    doi:10.1086/304599 , keywords =

    , volume = 487, number = 1, pages =. doi:10.1086/304599 , keywords =

    work page doi:10.1086/304599

  28. [28]

    doi:10.1142/S0218271814300079 , keywords =

    International Journal of Modern Physics D , volume = 23, number = 4, pages =. doi:10.1142/S0218271814300079 , keywords =

    doi:10.1142/s0218271814300079

  29. [29]

    doi:10.1103/PhysRevLett.22.728 , adsurl =

    , volume = 22, number = 14, pages =. doi:10.1103/PhysRevLett.22.728 , adsurl =

    work page doi:10.1103/physrevlett.22.728

  30. [30]

    , volume = 453, pages = 883, doi =

  31. [31]

    Astroparticle Physics , publisher =

    Transition from galactic to extragalactic cosmic rays , author =. Astroparticle Physics , publisher =. doi:10.1016/j.astropartphys.2012.09.007 , url =

    doi:10.1016/j.astropartphys.2012.09.007 2012

  32. [32]

    Annual Review of Nuclear and Particle Science , publisher =

    Cosmic-Ray Propagation and Interactions in the Galaxy , author =. Annual Review of Nuclear and Particle Science , publisher =. doi:10.1146/annurev.nucl.57.090506.123011 , url =

    arXiv

  33. [33]

    , volume = 125, pages =

  34. [34]

    Slatyer and Douglas P

    Meng Su and Tracy R. Slatyer and Douglas P. Finkbeiner , year = 2010, month =. The Astrophysical Journal , publisher =. doi:10.1088/0004-637x/724/2/1044 , url =

    doi:10.1088/0004-637x/724/2/1044 2010

  35. [35]

    doi:10.1038/s41586-020-2979-0 , keywords =

    , volume = 588, number = 7837, pages =. doi:10.1038/s41586-020-2979-0 , keywords =

    doi:10.1038/s41586-020-2979-0

  36. [36]

    Karen Yang and Mateusz Ruszkowski and Ellen G

    H.-Y. Karen Yang and Mateusz Ruszkowski and Ellen G. Zweibel , year = 2022, month =. Fermi and. Nature Astronomy , publisher =. doi:10.1038/s41550-022-01618-x , url =

    doi:10.1038/s41550-022-01618-x 2022

  37. [37]

    , volume = 756, number = 2, pages = 181, doi =

  38. [38]

    Bipolar Hypershell Galactic Center Starburst Model: Further Evidence from i

    Yoshiaki Sofue , year = 2000, month =. Bipolar Hypershell Galactic Center Starburst Model: Further Evidence from i. The Astrophysical Journal , publisher =. doi:10.1086/309297 , url =

    doi:10.1086/309297 2000

  39. [39]

    doi:10.1086/344573 , keywords =

    , volume = 582, number = 1, pages =. doi:10.1086/344573 , keywords =

    doi:10.1086/344573

  40. [40]

    Galaxies , volume = 6, number = 1, pages = 29, doi =

  41. [41]

    , volume = 894, number = 2, pages = 117, doi =

  42. [42]

    The Astrophysical Journal , publisher =

    Possible Counterpart Signal of the Fermi Bubbles at the Cosmic-Ray Positrons , author =. The Astrophysical Journal , publisher =. doi:10.3847/1538-4357/accb55 , url =

    work page doi:10.3847/1538-4357/accb55

  43. [43]

    doi:10.1093/mnrasl/slv098 , keywords =

    , volume = 453, number = 1, pages =. doi:10.1093/mnrasl/slv098 , keywords =

    doi:10.1093/mnrasl/slv098

  44. [44]

    , volume = 761, number = 2, pages = 185, doi =

  45. [45]

    doi:10.1051/0004-6361/201833683 , keywords =

    , volume = 619, pages =. doi:10.1051/0004-6361/201833683 , keywords =

    doi:10.1051/0004-6361/201833683

  46. [46]

    doi:10.1051/0004-6361/201220394 , keywords =

    , volume = 553, pages =. doi:10.1051/0004-6361/201220394 , keywords =

    work page doi:10.1051/0004-6361/201220394

  47. [47]

    40th COSPAR Scientific Assembly , volume = 40, pages =

  48. [48]

    Astronomy & Astrophysics , publisher =

    Cosmic-ray spectrum in the local Galaxy , author =. Astronomy & Astrophysics , publisher =. doi:10.1051/0004-6361/201731149 , url =

    doi:10.1051/0004-6361/201731149

  49. [49]

    , volume = 793, number = 1, pages = 64, doi =

  50. [50]

    Journal of Computational Physics , publisher =

    Preserving monotonicity in anisotropic diffusion , author =. Journal of Computational Physics , publisher =. doi:10.1016/j.jcp.2007.07.026 , url =

    doi:10.1016/j.jcp.2007.07.026 2007

  51. [51]

    Miller and Joel N

    Matthew J. Miller and Joel N. Bregman , year = 2016, month =. The Astrophysical Journal , publisher =. doi:10.3847/0004-637x/829/1/9 , url =

    doi:10.3847/0004-637x/829/1/9 2016

  52. [52]

    doi:10.1051/0004-6361/201936275 , keywords =

    , volume = 631, pages =. doi:10.1051/0004-6361/201936275 , keywords =

    work page doi:10.1051/0004-6361/201936275

  53. [53]

    doi:10.1051/0004-6361/201322858 , keywords =

    , volume = 563, pages =. doi:10.1051/0004-6361/201322858 , keywords =

    work page doi:10.1051/0004-6361/201322858

  54. [54]

    doi:10.1016/j.jcp.2005.02.017 , adsurl =

    Journal of Computational Physics , volume = 208, number = 1, pages =. doi:10.1016/j.jcp.2005.02.017 , adsurl =

    doi:10.1016/j.jcp.2005.02.017 2005

  55. [55]

    I: spectrum and chemical composition , author =

    Diffusive propagation of cosmic rays from supernova remnants in the Galaxy. I: spectrum and chemical composition , author =. Journal of Cosmology and Astroparticle Physics , publisher =. doi:10.1088/1475-7516/2012/01/010 , issn =

    doi:10.1088/1475-7516/2012/01/010 2012

  56. [56]

    High Energy Astrophysics , author =

  57. [57]

    doi:10.1086/305329 , keywords =

    , volume = 495, number = 2, pages =. doi:10.1086/305329 , keywords =

    doi:10.1086/305329

  58. [58]

    doi:10.1086/310975 , keywords =

    , volume = 489, number = 2, pages =. doi:10.1086/310975 , keywords =

    doi:10.1086/310975

  59. [59]

    doi:10.1103/RevModPhys.86.1 , keywords =

    Reviews of Modern Physics , volume = 86, number = 1, pages =. doi:10.1103/RevModPhys.86.1 , keywords =

    doi:10.1103/revmodphys.86.1

  60. [60]

    doi:10.1051/0004-6361/201117353 , archiveprefix =

    , volume = 538, pages =. doi:10.1051/0004-6361/201117353 , archiveprefix =. 1111.0962 , keywords =

    Pith/arXiv arXiv

  61. [61]

    , volume = 798, pages = 7, doi =

  62. [62]

    doi:10.1093/mnras/sty2857 , keywords =

    , volume = 483, number = 1, pages =. doi:10.1093/mnras/sty2857 , keywords =

    doi:10.1093/mnras/sty2857

  63. [63]

    , volume = 262, number = 2, pages = 36, doi =

  64. [64]

    doi:10.1086/152513 , adsurl =

    , volume = 186, pages =. doi:10.1086/152513 , adsurl =

    doi:10.1086/152513

  65. [65]

    doi:10.1093/mnrasl/slac146 , keywords =

    , volume = 519, number = 1, pages =. doi:10.1093/mnrasl/slac146 , keywords =

    work page doi:10.1093/mnrasl/slac146

  66. [66]

    doi:10.1086/340784 , keywords =

    , volume = 569, number = 2, pages =. doi:10.1086/340784 , keywords =

    doi:10.1086/340784

  67. [67]

    , volume = 152, number = 6, pages = 157, doi =

  68. [68]

    doi:10.1086/317148 , keywords =

    , volume = 543, number = 2, pages =. doi:10.1086/317148 , keywords =

    doi:10.1086/317148

  69. [69]

    doi:10.1093/mnras/213.3.451 , keywords =

    , volume = 213, pages =. doi:10.1093/mnras/213.3.451 , keywords =

    doi:10.1093/mnras/213.3.451

  70. [70]

    doi:10.1093/mnras/stab2553 , keywords =

    , volume = 508, number = 1, pages =. doi:10.1093/mnras/stab2553 , keywords =

    doi:10.1093/mnras/stab2553

  71. [71]

    doi:10.1103/PhysRevD.95.043541 , keywords =

    , volume = 95, number = 4, pages =. doi:10.1103/PhysRevD.95.043541 , keywords =

    doi:10.1103/physrevd.95.043541

  72. [72]

    doi:10.1103/PhysRevLett.126.071302 , keywords =

    , volume = 126, number = 7, pages =. doi:10.1103/PhysRevLett.126.071302 , keywords =

    doi:10.1103/physrevlett.126.071302

  73. [73]

    Computational Astrophysics and Cosmology , volume = 6, number = 1, pages = 2, doi =

  74. [74]

    doi:10.1093/mnras/stz2306 , keywords =

    , volume = 490, number = 3, pages =. doi:10.1093/mnras/stz2306 , keywords =

    doi:10.1093/mnras/stz2306

  75. [75]

    doi:10.1093/mnras/stz2338 , keywords =

    , volume = 490, number = 3, pages =. doi:10.1093/mnras/stz2338 , keywords =

    doi:10.1093/mnras/stz2338

  76. [76]

    A&A , volume = 361, pages =

  77. [77]

    doi:10.1086/184292 , keywords =

    , volume = 282, pages =. doi:10.1086/184292 , keywords =

    doi:10.1086/184292

  78. [78]

    doi:10.1093/mnras/stx3117 , keywords =

    , volume = 474, number = 4, pages =. doi:10.1093/mnras/stx3117 , keywords =

    doi:10.1093/mnras/stx3117

  79. [79]

    , volume = 904, number = 2, pages = 170, doi =

  80. [80]

    doi:10.1093/mnras/stac816 , keywords =

    , volume = 512, number = 4, pages =. doi:10.1093/mnras/stac816 , keywords =

    doi:10.1093/mnras/stac816

Showing first 80 references.