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REVIEW 3 major objections 4 minor 104 references

Finite upstream temperature monotonically suppresses the maximum deflection angle of relativistic oblique shocks, and in the ultra-thermal, ultra-relativistic limit the detachment angle becomes a universal function of the adiabatic index.

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2026-08-01 08:48 UTC pith:5U5SWK7U

load-bearing objection A clean analytic reformulation of relativistic oblique shocks, with a genuinely new finite-temperature correction and a universal angle that is less universal than advertised. the 3 major comments →

arxiv 2607.20978 v1 pith:5U5SWK7U submitted 2026-07-23 astro-ph.HE

Relativistic Oblique Shocks at Finite Temperature: Detachment Angle, Shock Polars, and the Turning Parameter

Rushikesh Ashok Sonkusale , Anshuman Verma , Ritam Mallick This is my paper
classification astro-ph.HE
keywords relativistic oblique shocksshock polarsdetachment angleturning parameterfinite temperatureTaub adiabatpulsar wind nebulashock thermodynamics
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 tries to establish that finite upstream temperature is not a small correction in relativistic oblique shocks: any nonzero thermal pressure lowers the maximum deflection angle, so cold-fluid models systematically overstate how easily a shock stays attached. A single dimensionless quantity, the turning parameter R, is introduced; it absorbs the equation of state, Mach number, and thermal state, and the entire shock polar follows from one compact relation. In the combined ultra-hot, ultra-relativistic limit, R saturates to Γ−1, and the detachment angle becomes a universal function of the adiabatic index alone: χ∞=arcsin[(2−Γ)/Γ]. The paper verifies these results numerically with shock polars and applies them to the Crab nebula, where Γ=4/3 gives χ∞≈30°, matching the observed torus extent.

Core claim

The central claim is that the turning parameter R=(h2/τ)/h1 is a sufficient statistic for relativistic oblique shock geometry. Since any finite upstream thermal content raises R above its cold value, and the turning relation sin(2φ−χ)=((1+R)/(1−R)) sinχ has a prefactor that grows monotonically with R, the shock polar necessarily shrinks and the maximum deflection angle χ_max falls; the authors prove this perturbatively to first order in the upstream thermal parameter. In the simultaneous ultra-thermal and ultra-relativistic limit, R saturates to Γ−1 independent of the shock angle, making the stationarity condition purely geometric and forcing the universal detachment angle χ∞=arcsin[(2−Γ)/Γ]

What carries the argument

The turning parameter R≡(h2/τ)/h1, the downstream specific enthalpy per unit compressed mass divided by the upstream enthalpy, is the load-bearing object. It collapses the equation of state, upstream Mach number, and thermal content into one scalar, so the turning relation sin(2φ−χ)=((1+R)/(1−R)) sinχ fully determines the shock polar. As R grows, the prefactor (1+R)/(1−R) grows, shrinking the polar; in the ultra-thermal limit the Taub adiabat forces ξ∼(Γ−1)τ² and hence R≈ξ/τ²→Γ−1, at which point ∂R/∂φ=0 and the stationarity condition reduces to cos(2φ−χ)=0, yielding the universal detachment angle.

Load-bearing premise

The derivation of the saturated turning parameter and universal detachment angle assumes a single polytropic index Γ holds all the way into the ultra-thermal, ultra-relativistic regime; if the equation of state changes—through pair production, radiation, or magnetization—before that limit, R∞=Γ−1 and χ∞=arcsin[(2−Γ)/Γ] no longer follow.

What would settle it

Take a relativistic oblique shock with M1=10, Γ=5/3, and a warm upstream state α1=0.1, and measure the shock-polar apex: the paper predicts χ_max is strictly smaller than the cold-limit value. Finding a larger apex would falsify the suppression claim. Independently, a Γ=4/3 ultra-hot simulation with M1≫1 should show the detachment angle hugging 30°; a value clearly above arcsin[(2−Γ)/Γ] would falsify the universal limit.

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

If this is right

  • Cold-fluid analyses overestimate the maximum deflection angle whenever the upstream plasma is warm, so detachment thresholds and bow-shock stand-off distances in supernova remnants, jets, and pulsar wind nebulae need downward revision.
  • At high temperature and high Mach number the detachment angle approaches χ∞=arcsin[(2−Γ)/Γ], giving a parameter-free way to infer the effective adiabatic index of a relativistic flow from its shock geometry.
  • Finite temperature lifts the cold Rankine–Hugoniot compression ceiling, so downstream compression can grow with Mach number; because synchrotron emissivity scales with compression, warm shocks can be far brighter than cold-limit estimates suggest.
  • The non-monotonic dependence of χ_max on Mach number at intermediate temperatures means that increasing a hot flow's bulk kinetic energy can first widen, then narrow, the allowed deflection—something cold-shock models cannot capture.
  • For a Γ=4/3 flow, the universal angle is 30°, and the paper's Crab application shows that finite temperature both suppresses the torus-width angle and brightens the inner ring, with flares producing a transient narrowing of the torus.

Where Pith is reading between the lines

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

  • If R is truly a sufficient statistic, then shock polars should collapse onto one another when different combinations of Mach number and temperature produce the same R; this is a direct, testable scaling prediction the paper does not state.
  • The ultra-thermal limit assumes a single polytropic index persists. In real plasmas, pair production and radiation will soften the equation of state before α1→∞, so the universal χ∞ may be approached but never exactly reached; the distance from χ∞ becomes a measure of how far the equation of state has changed.
  • Because the paper leaves magnetic fields out, a magnetized extension would likely need a second control parameter (magnetization); the turning parameter could still organize the polar, but χ∞ would probably acquire a magnetization dependence.
  • The Crab comparison is self-acknowledged illustrative; a sharper test would map the latitude-dependent detachment angle across the torus and fit the Gaussian thermal profile the paper assumes, checking whether the predicted χ_max(λ) shape matches the X-ray morphology.

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 / 4 minor

Summary. This paper develops a thermodynamic framework for relativistic oblique shocks with finite upstream temperature. The central object is the turning parameter R=(h2/τ)/h1. From the Taub adiabat and the conservation laws the authors derive the master implicit equation (Eq. 8), the turning relation (Eq. 11), and the detachment condition (Eq. 12). In the cold limit the standard Rankine-Hugoniot and Landau-Lifshitz relations are recovered. A first-order expansion in the upstream thermal parameter α1 yields corrections τ1, R1, and χ1 (Eqs. 16–17 and Appendix F), leading to the claim that finite upstream temperature raises R and lowers χmax. In the combined α1→∞, Mn→∞ limit the turning parameter saturates to R∞=Γ−1, giving χ∞=arcsin[(2−Γ)/Γ] (Eqs. 18–20) and recovering the Shi et al. minimum-intensity locus ϕ=χ/2+π/4. Numerical shock polars illustrate thermal suppression, non-monotonic χmax(M1), and convergence to χ∞. The paper concludes with an illustrative application to the Crab torus, where Γ=4/3 gives χ∞≈30°, compared with the observed λ≈29°.

Significance. If the results hold, the paper offers a compact single-parameter description of relativistic oblique-shock deflection, an analytic derivation of the Shi et al. locus, and a systematic first-order treatment of upstream temperature. Strengths include the transparent derivation from conservation laws, the exact recovery of the cold limit, the absence of fitted parameters in the core algebra, and the broad numerical shock-polar survey. The paper is also candid that the magnetized case is left to future work. However, the central sign statement R1>0 is not proved, the global monotonicity claim goes beyond the first-order analytical result, and the 'universal' asymptotic angle is conditional on a fixed polytropic index. These issues need to be addressed before the main claims are fully supported.

major comments (3)
  1. [Eq. (17) and Appendix F] The paper's central result χ1<0—and hence the statement that finite temperature suppresses the maximum deflection—rests on the assertion that R1>0 'for all physically admissible strong shocks' (Sec. II). Appendix F gives expressions for τ1, R1, and K, but it never proves R1>0; it only asserts it. K>0 is shown, but positivity of R1 is not automatic from Eq. (F1)–(F2), especially because of the denominator in τ1. Since this is load-bearing for the main claim, please supply a complete sign proof over the physical domain (Mn>1, 1<Γ<2), or replace the unconditional statement with a precise domain of validity and a numerical verification.
  2. [Abstract and Sec. II (Eq. 15); Sec. III (Figs. 3–4)] The abstract claims that 'any finite upstream temperature monotonically suppresses the maximum deflection angle.' The analytical support is first-order in α1 (Eq. 15 and Appendix F), and the numerical evidence is for selected parameters (Γ=5/3; a set of M1 values). No proof establishes global monotonicity in α1 for arbitrary M1 and Γ. The text in Sec. II similarly moves from a first-order perturbative result to 'any finite upstream thermal content increases the turning parameter' without a limiting statement. Please either prove the global statement or weaken the conclusion to 'to first order in α1, and in all computed numerical cases, ...'.
  3. [Appendix G and Sec. III.A] The universal saturation R∞=Γ−1 and χ∞=arcsin[(2−Γ)/Γ] are derived by taking α1→∞ and Mn→∞ while keeping a single polytropic index Γ fixed throughout. The authors are transparent about leaving magnetized and more general EOS to future work, but the abstract and Sec. III.A present χ∞ as 'depends only on the equation of state' and as a 'parameter-free prediction' for the Crab. In realistic relativistic plasmas Γ is not constant (pair production, radiation, magnetization), and the enthalpy relation changes, so Eq. (20) is conditional on an ideal polytrope and on the double limit being reached. The agreement with the observed Crab torus extent λ≈29° is therefore not a robust test of the model. Please qualify the universality/parameter-free wording and state explicitly the conditions under which Eq. (20) applies.
minor comments (4)
  1. [Fig. 3 caption] The caption says 'the quadratic onset of the thermal correction, χmax(α1)≃χ0+α1χ1'. This is inconsistent: Eq. (15) is linear in α1. Use 'linear onset' or clarify that the smallness of the coefficient makes the plateau appear flat.
  2. [Sec. III.A (flare paragraph)] The text says 'the increase in M1, which alone tends to increase χ∞'. Since χ∞ is independent of M1, this should read 'χmax'.
  3. [Sec. III (numerical validation)] The numerical shock-polar calculation solves the same master equation (Eq. 8) used to derive the analytics, so the agreement is primarily a consistency check rather than an independent validation. Consider stating this explicitly.
  4. [General notation] The normal Mach number is denoted M_n, M_{n1}, and M1 in different places (e.g., Eq. 9, Eq. 13, Fig. 6). Define the notation once and use it consistently.

Circularity Check

0 steps flagged

No significant circularity: R, R∞, and χ∞ are derived from the Rankine–Hugoniot/Taub equations; all inputs (α1, M1, Γ) are stated, and no fitted parameter is relabeled as a prediction.

full rationale

The derivation is self-contained. The central objects are defined from the EOS h_i = 1 + a p_i/ρ_i (Eq. 2) and the conservation laws (Eqs. 3–6); the Taub adiabat (Eq. 7) yields the master equation (Eq. 8). The turning relation (Eq. 11) is obtained from Eqs. (3) and (5) by algebra (Appendix C), not by inserting the desired χ_max. The first-order corrections τ1, R1, χ1 (Eqs. 16–17, F4) come from expanding this same master equation; no data are fitted, so nothing is a fitted input relabeled as prediction. The ultra-thermal result R∞ = Γ−1 (Eq. 18) follows in Appendix G from the asymptotic Taub relation with rest mass negligible and the requirement that the equation remain finite as M_n → ∞; the universal χ∞ (Eq. 20) is then evaluated from the turning relation. This is a limiting consequence, not a restatement of the input. The recovery of the Shi et al. locus is a derivation from R-saturation, with the earlier geometric result cited only as the known fact being explained. The numerical shock polars solve the same master equation used for the analytics, so they are a consistency check rather than an independent empirical validation, but this does not constitute circularity because the analytics are not derived from the numerics. The authors flag the magnetized-Crab treatment as illustrative and future work (Secs. III A, IV, G), and the assumption of a constant Γ is an EOS-robustness caveat, not a circular step. Self-citations (e.g., Refs. 12–18) are contextual applications and carry none of the derivation's weight.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central derivation has no fitted parameters: α1, M1, and Γ are physical inputs and R is derived. The only hand-chosen parameters appear in the illustrative Crab application (α1eq, σ, flare profile values). The 'parameter-free' Crab prediction still requires choosing Γ=4/3, so that label is softer than stated. The main axioms are the perfect-fluid assumption, constant-Γ polytropic EOS, and the double-limit ordering for the universal angle.

free parameters (3)
  • equatorial thermal parameter α1eq = 0.1 (strong), 0.05 (moderate)
    Chosen by hand in Eq. (21) to represent striped-wind reconnection heating; not fitted to the Crab torus; it sets the suppression amplitude in Figs. 7–9.
  • latitude width σ = 20°
    Chosen in Eq. (21) as representative of the striped-wind reconnection zone; affects the latitude profile of χmax and the emissivity proxy.
  • flare profile parameters = M_base=50, M_peak=300, α1,base=0.05, α1,peak=0.5, σ_t=1.2
    Hand-set in Eqs. (22)–(23) to mimic a Crab flare; illustrative only, not constrained by the data.
axioms (5)
  • domain assumption Perfect-fluid relativistic hydrodynamics: T^{μν}=ρh u^μ u^ν + p g^{μν}, baryon conservation, and energy-momentum conservation across the shock (Eqs. 3–6).
    Foundation of all shock jump conditions; standard for the regime but excludes magnetic fields, viscosity, and heat conduction.
  • domain assumption Polytropic equation of state with constant Γ: h_i = 1 + Γ/(Γ−1) p_i/ρ_i (Eq. 2).
    Used in the master equation, the perturbative expansion, and the R∞/χ∞ limits. If Γ varies across the shock, the formulas do not follow.
  • domain assumption Upstream thermal parameter α1 = p1/ρ1 = kT/m (Sec. II).
    Connects temperature to pressure; assumes an ideal gas and a single particle species.
  • standard math The detachment point is identified by F=0 and ∂F/∂φ=0 (Eq. 12).
    Standard coalescence of the weak and strong shock branches; used to define χmax.
  • domain assumption In the ultra-thermal limit rest-mass terms are negligible, and Mn→∞ forces ξ→∞ so that the coefficient of ξ in Eq. (G2) must vanish.
    This double-limit ordering is what produces R∞=Γ−1 and hence χ∞; it is an assumption about which limit is reached before other microphysics intervenes.

pith-pipeline@v1.3.0-alltime-deepseek · 16454 in / 14174 out tokens · 150934 ms · 2026-08-01T08:48:33.214747+00:00 · methodology

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read the original abstract

Oblique shocks are ubiquitous in high-energy astrophysical environments, yet a systematic analytical treatment of how finite upstream temperature influences the maximum deflection angle has been lacking. We address this problem by developing a unified thermodynamic framework based on a novel dimensionless quantity, the turning parameter, which encapsulates the equation of state, upstream Mach number, and thermal state of the flow into a single variable. Starting from the relativistic Rankine-Hugoniot conditions and the Taub adiabat, we derive a compact turning relation and a first-order perturbative expansion in the upstream thermal parameter. We show that any finite upstream temperature monotonically suppresses the maximum deflection angle relative to the cold-fluid limit, implying that cold models systematically overestimate shock attachment. In the combined ultra-thermal and ultra-relativistic limit, the turning parameter saturates to a universal value, yielding an asymptotic detachment angle that depends only on the equation of state. Numerical shock-polar calculations validate the analytical results and reveal a non-monotonic dependence of the detachment angle on the Mach number at intermediate temperatures, arising from the competition between thermal pressure and bulk kinetic energy-a distinctly relativistic thermal effect absent in both the cold and ultra-hot limits. As an illustrative astrophysical application, we apply the framework to the Crab pulsar wind nebula, demonstrating how finite-temperature effects modify the termination-shock morphology and the observed torus geometry.

Figures

Figures reproduced from arXiv: 2607.20978 by Anshuman Verma, Ritam Mallick, Rushikesh Ashok Sonkusale.

Figure 1
Figure 1. Figure 1: FIG. 1. Shock geometry in the shock rest frame. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Shock-polar evolution with upstream Mach number (Γ = 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: increasing either α1 or M1 raises R toward its asymptotic value R∞ = Γ−1, while only their combined large-limit recovers the fully saturated thermodynamic regime predicted analytically in Sec. G. Taken together, Figs. 2–6 demonstrate that the turn￾ing parameter R is not merely an auxiliary quantity but rather the fundamental thermodynamic variable that governs the entire family of shock solutions. The non￾… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Maximum deflection angle [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Convergence of the turning parameter [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Transient flare evolution. Top panel shows the up [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Equatorial density spike from thermal ceiling break [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Shock polar ( [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗

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Reference graph

Works this paper leans on

104 extracted references · 2 linked inside Pith

  1. [1]

    Compression Correctionτ 1 Expanding the terms in the master equation (B1) to O(α2

  2. [2]

    and isolating theO(α 2

  3. [3]

    Grouping theτ 1 terms and simplifying yields the compression cor- rection: τ1 = aτ 2 0 (ξ−1 +a) +aξ[(ξ−1)−aξ] (ξ−1)−2aξ ,(F1) whereξ=ξ cold at zeroth order

    coefficients requires ex- pandingh 2 2 −h 2 1 against the right-hand side. Grouping theτ 1 terms and simplifying yields the compression cor- rection: τ1 = aτ 2 0 (ξ−1 +a) +aξ[(ξ−1)−aξ] (ξ−1)−2aξ ,(F1) whereξ=ξ cold at zeroth order. 11

  4. [4]

    T urning-Parameter CorrectionR1 ExpandingR= (h 2/τ)/h 1 to first order using the ex- pansion of 1/τandh −1 1 = 1−aα 1 +O(α 2

  5. [5]

    gives: R1 = aξ−τ 1 τ 2 0 − a τ0 .(F2)

  6. [6]

    At the cold detachment point (ϕ 0, χ0, R0),G= 0 and ∂G/∂ϕ= 0

    Detachment-Angle Correctionχ 1 DefineG(ϕ, χ, R)≡sin(2ϕ−χ)−[(1+R)/(1−R)] sinχ. At the cold detachment point (ϕ 0, χ0, R0),G= 0 and ∂G/∂ϕ= 0. Settingϕ=ϕ 0 +α 1ϕ1,χ=χ 0 +α 1χ1, and R=R 0 +α 1R1, a Taylor expansion toO(α 1) gives: ∂G ∂χ 0 χ1 + ∂G ∂R 0 R1 = 0,(F3) since theϕ 1 term vanishes via the stationarity condi- tion. Evaluating the partial derivatives a...

  7. [7]

    Saturated T urning ParameterR∞ Forα 1 → ∞, rest-mass terms become negligible (h1 ≈ aα1,h 2 ≈aξα 1/τ). Theα 1 factors cancel in the turning parameterR: R≈ ξ τ 2 ≡Y.(G1) Substituting these approximations into the Taub adia- bat (A4) and collecting terms inξyields: ξ(aY−1−Y) =a−1−Y.(G2) Taking the limitM n1 → ∞forces the pressure ratioξ→ ∞. For (G2) to remai...

  8. [8]

    Universal Detachment Angleχ ∞ WithR= Γ−1 = const, the derivative∂R/∂ϕvan- ishes. The stationarity condition (D2) reduces purely to geometry (cos(2ϕ−χ) = 0), yielding the minimum- intensity locus: ϕ= χ 2 + π 4 .(G4) Substituting this locus andR= Γ−1 into the turning relation (C5) gives sin(π/2) = [Γ/(2−Γ)] sinχ ∞. Solving forχ ∞ gives the universal detachm...

  9. [9]

    = 4 (horizontal dashed) is progressively lifted; hotter gas sustains unbounded growthτ∝M n1. The gradual bending of each curve away from the cold ceiling marks the Mach number above which thermal enthalpy generation, rather than compressional heating, dominates the downstream state

  10. [10]

    A. M. Anile,Relativistic Fluids and Magneto-Fluids (Cambridge University Press, Cambridge, 1989)

    1989

  11. [11]

    L. D. Landau and E. M. Lifshitz,Fluid Mechanics, 2nd ed. (Pergamon Press, Oxford, 1987)

    1987

  12. [12]

    R. A. Chevalier, Astrophys. J.258, 790 (1982)

    1982

  13. [13]

    S. F. Kamijima and Y. Ohira, Physical Review D110, 043046 (2024)

    2024

  14. [14]

    Petruk, R

    O. Petruk, R. Bandiera, T. Kuzyo, R. Brose, and A. Ingallinera, Astronomy & Astrophysics 10.1051/0004- 6361/202557311 (2026)

    doi:10.1051/0004- 2026

  15. [15]

    Cristofari, Astronomy & Astrophysics704, A213 (2025)

    P. Cristofari, Astronomy & Astrophysics704, A213 (2025)

    2025

  16. [16]

    R. D. Blandford and M. J. Rees, Mon. Not. R. Astron. Soc.169, 395 (1974)

    1974

  17. [17]

    Daigne and R

    F. Daigne and R. Mochkovitch, Mon. Not. R. Astron. Soc.296, 275 (1998)

    1998

  18. [18]

    Murase and S

    K. Murase and S. Nagataki, Phys. Rev. D73, 063002 (2006)

    2006

  19. [19]

    Foffano and M

    L. Foffano and M. Tavani, The Astrophysical Journal 991, 82 (2025)

    2025

  20. [20]

    Charlet, J

    A. Charlet, J. Granot, and P. Beniamini, Astronomy & Astrophysics699, A139 (2025)

    2025

  21. [21]

    Bhattacharyya, S

    A. Bhattacharyya, S. K. Ghosh, P. S. Joarder, R. Mallick, and S. Raha, Phys. Rev. C74, 065804 (2006)

    2006

  22. [22]

    Mishustin, R

    I. Mishustin, R. Mallick, R. Nandi, and L. Satarov, Phys. Rev. C91, 055806 (2015)

    2015

  23. [23]

    Prasad and R

    R. Prasad and R. Mallick, Astrophys. J.859, 57 (2018)

    2018

  24. [24]

    Mallick and M

    R. Mallick and M. Irfan, Mon. Not. R. Astron. Soc.485, 577 (2019)

    2019

  25. [25]

    Mallick, S

    R. Mallick, S. Singh, and R. Prasad, Mon. Not. R. As- tron. Soc.507, 1318 (2021)

    2021

  26. [26]

    Mallick and A

    R. Mallick and A. Verma, J. High Energy Astrophys.36, 36 (2022)

    2022

  27. [27]

    Verma and R

    A. Verma and R. Mallick, Mon. Not. R. Astron. Soc.522, 4801 (2023)

    2023

  28. [28]

    L. P. Csernai and M. Gong, Phys. Rev. D37, 3231 (1988)

    1988

  29. [29]

    Cissoko, Phys

    M. Cissoko, Phys. Rev. D45, 1045 (1992)

    1992

  30. [30]

    Senno, K

    N. Senno, K. Murase, and P. M´ esz´ aros, Phys. Rev. D93, 083003 (2016)

    2016

  31. [31]

    Peretti, E

    E. Peretti, E. Amato, S. S. Cerri, G. Morlino, L. Per- fetta Pullano, and S. Recchia, arXiv e-prints (2026), arXiv:2603.16647 [astro-ph.HE]

    arXiv 2026

  32. [32]

    Saiz-P´ erez, C

    A. Saiz-P´ erez, C. M. Fromm, M. Perucho, O. Porth, M. Kadler, Y. Mizuno, A. Chael, and K. Mannheim, As- tronomy & Astrophysics693, A169 (2025)

    2025

  33. [33]

    Mallick and S

    R. Mallick and S. Schramm, Phys. Rev. C89, 045805 (2014)

    2014

  34. [34]

    Prasad and R

    R. Prasad and R. Mallick, Astrophys. J.893, 151 (2020)

    2020

  35. [35]

    L. O. Drury, Rep. Prog. Phys.46, 973 (1983)

    1983

  36. [36]

    Courant and K

    R. Courant and K. O. Friedrichs,Supersonic Flow and Shock Waves(Interscience Publishers, New York, 1948)

    1948

  37. [37]

    R. D. Blandford and C. F. McKee, Phys. Fluids19, 1130 (1976)

    1976

  38. [38]

    R. B. Decker and L. Vlahos, Journal of Geophysical Re- search90, 47 (1985)

    1985

  39. [39]

    R. B. Decker, S. M. Krimigis, E. C. Roelof, M. E. Hill, T. P. Armstrong, G. Gloeckler, D. C. Hamilton, and L. J. Lanzerotti, Nature454, 67 (2008)

    2008

  40. [40]

    Huang, B

    Z.-Q. Huang, B. Reville, J. G. Kirk, and G. Giacinti, Monthly Notices of the Royal Astronomical Society522, 4955 (2023)

    2023

  41. [41]

    Lemoine, V

    M. Lemoine, V. Bresci, and L. Gremillet, Physical Re- view D112, 123028 (2025)

    2025

  42. [42]

    A. H. Taub, Phys. Rev.74, 328 (1948)

    1948

  43. [43]

    Rezzolla and O

    L. Rezzolla and O. Zanotti,Relativistic Hydrodynamics (Oxford University Press, Oxford, 2013). 13

    2013

  44. [44]

    A. Shi, J. Chen, E. H. Dowell, H. Wen, X. Li, and T. Shen, Astrophys. J.893, 167 (2020)

    2020

  45. [45]

    M. H. Johnson and C. F. McKee, Phys. Rev. D3, 858 (1971)

    1971

  46. [46]

    C. F. Kennel and F. V. Coroniti, Astrophys. J.283, 694 (1984)

    1984

  47. [47]

    J. G. Kirk, Y. Lyubarsky, and J. Petri, Astrophys. Space Sci. Libr.357, 421 (2009)

    2009

  48. [48]

    Svensson, Astrophys

    R. Svensson, Astrophys. J.258, 335 (1982)

    1982

  49. [49]

    J. A. S. Lima, A. Kandus, and R. Opher, Phys. Rev. D 67, 023002 (2003)

    2003

  50. [50]

    B. M. Gaensler, J. Arons, V. M. Kaspi, M. J. Pivovaroff, N. Kawai, and K. Tamura, Astrophys. J.569, 878 (2002)

    2002

  51. [51]

    B¨ uhler and M

    R. B¨ uhler and M. Giomi, Monthly Notices of the Royal Astronomical Society462, 2762 (2016)

    2016

  52. [52]

    Porth, R

    O. Porth, R. B¨ uhler, B. Olmi, S. Komissarov, A. Lam- berts, E. Amato, Y. Yuan, and A. Rudy, Space Science Reviews207, 137 (2017)

    2017

  53. [53]

    J. T. Dinsmore and R. W. Romani, The Astrophysical Journal976, 4 (2024)

    2024

  54. [54]

    B. Olmi, E. Amato, R. Bandiera, and P. Blasi, arXiv e-prints (2024), arXiv:2403.03616 [astro-ph.HE]

    Pith/arXiv arXiv 2024

  55. [55]

    M. V. Barkov, M. Lyutikov, N. Klingler, and P. Bordas, Monthly Notices of the Royal Astronomical Society485, 2041 (2019)

    2041

  56. [56]

    O. D. Toropina, M. M. Romanova, and R. V. E. Lovelace, Monthly Notices of the Royal Astronomical Society484, 1475 (2019)

    2019

  57. [57]

    Posselt, G

    B. Posselt, G. G. Pavlov, P. O. Slane, R. Romani, N. Buc- ciantini, A. M. Bykov, O. Kargaltsev, M. C. Weisskopf, and C.-Y. Ng, The Astrophysical Journal835, 66 (2017)

    2017

  58. [58]

    Gagnon, O

    S. Gagnon, O. Kargaltsev, J. Alford, J. Gelfand, and A. Lange, arXiv e-prints (2026), arXiv:2605.21278 [astro- ph.HE]

    Pith/arXiv arXiv 2026

  59. [59]

    Caoet al., Science373, 425 (2021)

    Z. Caoet al., Science373, 425 (2021)

    2021

  60. [60]

    Caoet al., The Astrophysical Journal Supplement Se- ries271, 25 (2024)

    Z. Caoet al., The Astrophysical Journal Supplement Se- ries271, 25 (2024)

    2024

  61. [61]

    Li, X.-J

    J. Li, X.-J. Bi, L.-Q. Gao, X. Huang, R.-M. Yao, and P.-F. Yin, Chinese Physics C48, 065107 (2024)

    2024

  62. [62]

    Alispachet al., Astronomy & Astrophysics699, A255 (2025)

    C. Alispachet al., Astronomy & Astrophysics699, A255 (2025)

    2025

  63. [63]

    S. T. Spencer, A. M. W. Mitchell, and B. Reville, As- tronomy & Astrophysics698, A131 (2025)

    2025

  64. [64]

    Aharonianet al., Astronomy & Astrophysics686, A308 (2024)

    F. Aharonianet al., Astronomy & Astrophysics686, A308 (2024)

    2024

  65. [65]

    W. Wei, F. Xie, F. La Monaca, W. Deng, M. Ge, K. Liu, C. Zuo, and W. Chen, Monthly Notices of the Royal As- tronomical Society539, 902 (2025)

    2025

  66. [66]

    C. Zuo, F. Xie, M. Ge, W. Deng, K. Liu, F. La Monaca, A. Di Marco, W. Wei, and W. Chen, The Astrophysical Journal983, 1 (2025)

    2025

  67. [67]

    Martin, D

    T. Martin, D. Milisavljevic, T. Temim, S. Mandal, P. Duffell, L. Drissen, and Z. Ding, Monthly Notices of the Royal Astronomical Society544, 1 (2025)

    2025

  68. [68]

    Chatterjee, H.-K

    D. Chatterjee, H.-K. Chang, D. Debnath, K. A. Augus- tine, and T.-H. Lin, The Astrophysical Journal988, 105 (2025)

    2025

  69. [69]

    Temim, J

    T. Temim, J. M. Laming, P. J. Kavanagh, N. Smith, P. Slane, W. P. Blair, I. De Looze, N. Bucciantini, A. Jerkstrand, N. M. Gountanis, R. Sankrit, D. Mil- isavljevic, A. Rest, M. Lyutikov, J. DePasquale, T. Mar- tin, L. Drissen, J. Raymond, O. D. Fox, M. Modjaz, A. Spitkovsky, and L. G. Strolger, The Astrophysical Journal Letters968, L18 (2024)

    2024

  70. [70]

    B¨ uhler and R

    R. B¨ uhler and R. D. Blandford, Rep. Prog. Phys.77, 066901 (2014)

    2014

  71. [71]

    Porth, S

    O. Porth, S. S. Komissarov, and R. Keppens, Mon. Not. R. Astron. Soc.438, 278 (2014)

    2014

  72. [72]

    N. F. Camus, S. S. Komissarov, N. Bucciantini, and P. A. Hughes, Mon. Not. R. Astron. Soc.400, 1241 (2009)

    2009

  73. [73]

    Lyutikov, S

    M. Lyutikov, S. Komissarov, L. Sironi, and O. Porth, J. Plasma Phys.84, 635840201 (2018)

    2018

  74. [74]

    Cerutti, G

    B. Cerutti, G. R. Werner, D. A. Uzdensky, and M. C. Begelman, Astrophys. J.782, 104 (2014)

    2014

  75. [75]

    Cerutti, A

    B. Cerutti, A. A. Philippov, and G. Dubus, Astron. As- trophys.642, A204 (2020)

    2020

  76. [76]

    S. S. Komissarov and Y. E. Lyubarsky, Mon. Not. R. Astron. Soc.344, L93 (2003)

    2003

  77. [77]

    J. J. Hester, P. A. Scowen, R. Sankrit, C. J. Burrows, J. S. Gallagher, J. A. Holtzman, A. Watson, J. T. Trauger, G. E. Ballester, S. Casertano, J. T. Clarke, D. Crisp, R. W. Evans, R. E. Griffiths, J. G. Hoessel, J. Krist, J. R. Mould, K. R. Stapelfeldt, and J. A. Westphal, Astrophys. J.448, 240 (1995)

    1995

  78. [78]

    J. J. Hester, Annu. Rev. Astron. Astrophys.46, 127 (2008)

    2008

  79. [79]

    M. C. Weisskopf, J. J. Hester, A. F. Tennant, R. F. El- sner, N. S. Schulz, H. L. Marshall, M. Karovska, J. S. Nichols, D. A. Swartz, J. J. Kolodziejczak, and S. L. O’Dell, Astrophys. J. Lett.536, L81 (2000)

    2000

  80. [80]

    Ng and R

    C.-Y. Ng and R. W. Romani, Astrophys. J.601, 479 (2004)

    2004

Showing first 80 references.