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Treating equipartition corrections as independent underestimates synchrotron outflow energies by a factor of about five.

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2026-07-10 21:41 UTC pith:DT2WEU5H

load-bearing objection Solid, usable upgrade to equipartition analysis: self-consistent algebra plus public code that systematically raises energies by factors of a few. the 2 major comments →

arxiv 2607.06770 v1 pith:DT2WEU5H submitted 2026-07-07 astro-ph.HE

A Self-Consistent Framework for Synchrotron Equipartition Analysis

Coleman Rohde , Tanmoy Laskar , Noah Franz , Gavin Farley , Collin Christy , Kate D. Alexander This is my paper
classification astro-ph.HE
keywords synchrotron equipartitionrelativistic jetstidal disruption eventsradio transientsself-absorptionhot protonsoutflow energy
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.

Astronomers often use equipartition analysis to convert radio spectra into estimates of the energy, size, and speed of outflows that produce synchrotron emission. Earlier refinements for self-absorption, hot protons, and departures from strict equipartition were usually multiplied together after the fact, which produces inconsistent equations. This paper derives a single, closed framework in which those corrections depend on one another and are solved together for Newtonian outflows and for both on-axis and off-axis relativistic jets. When the full set is applied to well-studied tidal disruption events, a fast X-ray transient, and an active galactic nucleus, the inferred energies rise by factors of roughly five to six relative to earlier calculations that treated the corrections separately. The authors release open-source code so the same self-consistent analysis can be repeated for any synchrotron source. The result matters because energy is the primary observational handle on how these outflows are launched and how much power they deposit into their surroundings.

Core claim

When the energy of self-absorption-suppressed electrons, the energy stored in non-radiating particles, and the degree of departure from equipartition are allowed to enter the same minimization, the resulting minimum energy is systematically higher—by a factor of order five for typical microphysical parameters—than the value obtained by applying the same corrections independently.

What carries the argument

A self-consistent equipartition radius and energy in which the hot-proton parameter ξ ≡ (1 − ε_B)/ε_e multiplies the electron energy, the out-of-equipartition parameter ε incorporates ξ, and both appear inside the same algebraic minimizer for Newtonian and relativistic geometries.

Load-bearing premise

All energy components—radiating electrons, magnetic field, and non-radiating particles—are assumed to occupy exactly the same volume, so their energy fractions simply add to one.

What would settle it

Re-analyze a source whose radius or magnetic field is independently known (for example from resolved imaging or a measured cooling break) both with and without the coupled corrections; if the coupled solution systematically overshoots the independent constraint while the older independent-correction solution does not, the claimed energy boost is ruled out.

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

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

2 major / 5 minor

Summary. The manuscript derives a self-consistent equipartition formalism for Newtonian outflows and on- and off-axis relativistic jets that simultaneously incorporates SSA-suppressed electrons at ν_m, non-radiating energy (parameterized by ξ ≡ (1-ε_B)/ε_e), deviations from equipartition (via a redefined ε that includes ξ), and the exact numerical prefactor C that matches Rayleigh–Jeans and synchrotron peaks. The resulting expressions for R_eq, E_eq, and the 4-velocity constraint (eqs. 16–43, 48–55) recover the analytic limits of Barniol Duran et al. (2013), Matsumoto & Piran (2023), and related works when the corrections are switched off. A public code implements the full set of equations; applications to ASASSN-19bt, AT2019dsg, EP240414a, and J0231-0433 show that the interdependence of the corrections systematically raises equipartition energies by factors of a few (∼5 for the TDEs, ∼6 for EP240414a) relative to earlier analyses that applied the same corrections independently.

Significance. If the algebraic interdependence is correct, literature equipartition energies for many radio transients and AGN are systematically low by factors of several under standard microphysical parameters. That is a practically important correction for launch-mechanism constraints. Strengths that raise confidence include: (i) explicit recovery of prior analytic limits when corrections are disabled, (ii) a carefully bounded numerical root finder for the 4-velocity that does not assume Γ ≫ 1 or θ ≪ 1, (iii) a publicly released code, and (iv) transparent re-processing of published F_p, ν_p values with error propagation. The work is therefore a useful methodological advance for the community even if the absolute energy boost remains model-dependent.

major comments (2)
  1. Section 2.2 and eqs. (15)–(17): the entire energy-minimization argument assumes that non-thermal electrons, magnetic field, and ‘energy elsewhere’ occupy exactly the same volume V, so that ε_e + ε_o + ε_B = 1 and ξ = (1-ε_B)/ε_e. This is standard but load-bearing for the claimed factor-of-∼5 boost. The manuscript should state more explicitly how R_eq and E_eq change if the non-radiating component occupies a different volume (or provide a short appendix with the modified minimization), so that readers can judge the robustness of the numerical factor.
  2. Section 5.1 (ASASSN-19bt, off-axis models) and the paragraph following Table 1: when ν_m < ν_a is assumed, the derived γ_m exceeds γ_e = γ_a, which is inconsistent with the assumed spectral ordering; the authors set κ = 1 and note that the off-axis solutions may be unphysical. Because the abstract and conclusion advertise a factor-of-∼5 energy increase that is driven largely by these off-axis cases, the manuscript should either (i) present a parallel ν_a < ν_m analysis for the same epochs or (ii) clearly flag that the ∼5 factor for ASASSN-19bt is model-dependent and that the Newtonian solution is preferred under the stated assumptions.
minor comments (5)
  1. Eq. (7) and the accompanying footnote: C is discontinuous across ν_m = ν_a. A short remark on how the code handles the transition (or a plot of the jump) would help users avoid spurious discontinuities when both breaks are measured.
  2. Figure 2 caption and Table 1: the cosmology and microphysical parameters are stated, but the precise values of p used for each epoch of ASASSN-19bt are not listed in the table; adding them would improve reproducibility.
  3. Section 3, eq. (44): the Newtonian γ_m expression uses the shock-jump factor 9/32. A one-sentence citation or derivation of that factor would aid non-specialist readers.
  4. Throughout: a few typographical issues remain (“F ramework” in the title block, occasional missing spaces around ∼, and “ASSASN-19bt” once in §5.1). These are easily fixed.
  5. Section 5.3 (EP240414a): the large uncertainty on p produces very wide posteriors on N_e and n_ext. Quoting 75 % CIs is fine, but a brief note that these quantities are prior-dominated under the present sampling would be useful.

Circularity Check

0 steps flagged

No significant circularity: first-principles re-derivation of equipartition with interdependent corrections; applications re-ingest published peak fluxes and recover prior results when corrections are disabled.

full rationale

The paper derives R_eq, E_eq, and related quantities from synchrotron/Rayleigh-Jeans matching (eqs. 2–5), energy expressions (11–12), the definition ξ ≡ (1−ε_B)/ε_e (15), and minimization of E_tot = ξ E_e + E_B (16–21), then extends the same algebra to DFE (32–43), Newtonian γ_m (44–45), and dual-break cases (46–55). These steps are algebraic identities under the stated assumptions (shared volume V, power-law electrons, filling factors, etc.); none is defined in terms of the final numerical factor-of-~5 boost. Applications simply insert previously published F_p, ν_p (and, for J0231-0433, an independent angular size) and compare against earlier analyses; when the new corrections are switched off the code recovers Matsumoto & Piran (2023) and Barniol Duran et al. (2013) results, confirming that the boost is produced by the interdependence rather than by a fitted parameter renamed as a prediction. The only mild self-reference is the authors’ own earlier observational papers (Christy et al. 2024, Cendes et al. 2021) that supply the input SEDs; those citations are data sources, not load-bearing uniqueness theorems or ansatzes. The shared-volume assumption for ξ is standard and already flagged by the authors; it does not render the derivation circular. Score 1 reflects only that trivial self-citation of input data.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 1 invented entities

The central claim rests on standard synchrotron theory plus a small set of domain assumptions about volume sharing and microphysical fractions; no new particles or forces are postulated. The free parameters are the usual microphysical and geometric quantities that every equipartition analysis must choose; their typical values drive the numerical size of the energy boost but are not fitted to the paper’s own data.

free parameters (5)
  • ε_e (electron energy fraction) = 0.1 (assumed)
    Set to the conventional value 0.1 in all applications; enters ξ and the modified DFE parameter and therefore controls the size of the energy boost.
  • ε_B (magnetic energy fraction) = 0.003–0.2 (case-dependent)
    Either assumed (0.02, 0.003, 0.005, 0.2) or solved for from an independent radius measurement (J0231-0433); when ≪1 it amplifies the out-of-equipartition correction.
  • f_A, f_V, f_Ω (area/volume/solid-angle filling factors) = 1 or 0.36/4
    Geometric parameters set to 1 or 0.36/4 following earlier papers; they rescale R_eq and n_ext.
  • p (electron power-law index) = 2.16–2.8
    Taken from spectral fits of the input SEDs (p≈2.16–2.8); appears in every exponent of the equipartition expressions.
  • θ (observer angle for off-axis jets) = 0–1.57 rad
    Fixed to literature values (0.79–1.57 rad) when relativistic solutions are explored; controls which root of the 4-velocity equation is physical.
axioms (5)
  • domain assumption Total energy E = ξ E_e + E_B can be minimized with respect to radius to obtain a unique equipartition radius and energy (Section 2.3).
    Classic equipartition premise; the paper’s novelty is only in the self-consistent definition of ξ and the DFE parameter inside that minimization.
  • domain assumption All energy components occupy the identical volume V so that ε_e + ε_o + ε_B = 1 and ξ ≡ (1 − ε_B)/ε_e (Section 2.2).
    Required to close the energy budget; if the non-radiating component fills a different volume the algebraic structure changes.
  • domain assumption The observer-frame time–radius relation t = (1+z) R / [c β (1 − β cos θ)] holds and can be used to constrain Γ (eq. 26).
    Standard for constant-velocity or coasting outflows; mildly relativistic or decelerating cases would require a different dynamical mapping.
  • domain assumption γ_m = μ χ_e (Γ − 1) (relativistic) or the Newtonian shock-jump expression (eq. 44), with a floor at γ_m = 2.
    Standard shock-acceleration result; the floor is an ad-hoc but conventional cutoff when the formula yields sub-relativistic electrons.
  • domain assumption Numerical prefactors in the synchrotron peak-frequency and peak-flux formulas may be replaced by p-independent constants (or the exact C(p) of Shen & Zhang 2009) without changing the leading-order scalings.
    Explicitly stated approximation following Barniol Duran et al. and Matsumoto & Piran; the paper improves on it by retaining the full C(p) but still drops other p-dependent O(1) factors.
invented entities (1)
  • Self-consistent DFE parameter ε ≡ [11 / 2(p+1)] (ε_B / (ξ ε_e)) no independent evidence
    purpose: Quantifies deviation from equipartition while automatically incorporating the hot-proton correction ξ so that the two effects are no longer multiplied independently.
    Definitional construct introduced in Section 2.5; it has no independent existence outside the equipartition algebra and is therefore listed for completeness.

pith-pipeline@v1.1.0-grok45 · 28082 in / 3818 out tokens · 66903 ms · 2026-07-10T21:41:19.382629+00:00 · methodology

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read the original abstract

Determining the energy, size, and velocity of synchrotron-emitting outflows is essential for testing models of their formation and evolution, but these quantities are often poorly constrained by observations alone. Equipartition analysis, therefore, provides a widely used framework for estimating these properties. Prior works have developed refinements to account for additional physical effects and other sources of energy (e.g., self-absorption, hot protons, and deviations from strict equipartition); however, these corrections are typically applied independently of one another, resulting in internal inconsistencies. In this work, we derive a self-consistent equipartition framework that accounts for the interdependence of various correction factors for Newtonian outflows and on- and off-axis relativistic jets. We implement our framework in an easy-to-use, publicly available code and apply it to study the tidal disruption events ASASSN-19bt and AT2019dsg, fast X-ray transient EP240414a, and active galactic nucleus J0231-0433. The interdependence of the corrections can increase energy estimates by a factor of ~5, suggesting that the energies of other synchrotron sources may be similarly underestimated in the literature. These results indicate that simultaneously incorporating these correction factors is essential for determining accurate outflow properties and constraining launch mechanisms.

Figures

Figures reproduced from arXiv: 2607.06770 by Coleman Rohde, Collin Christy, Gavin Farley, Kate D. Alexander, Noah Franz, Tanmoy Laskar.

Figure 1
Figure 1. Figure 1: The on axis solution (circle), bounded below by uon,lb and above by umin, and the off-axis solution (square), bounded below by umin and above by uoff,ub, exist for (approximate) critical angles ˜θc ≈ θc larger than the observer angle θ i.e., for ˜θc ≳ θ. θ = 0.2 is fixed here as an example. Using equation 18 and in equipartition (R = Req), this yields the constraint δD =  β βeq,N − 2p+13 3(p+6) . (28) Th… view at source ↗
Figure 2
Figure 2. Figure 2: The ratio between equipartition quantities for the radio SEDs of the TDE ASASSN 19bt derived from our analysis and those reported by C. T. Christy et al. (2024) (which used the R. Barniol Duran et al. (2013, BNP13) formalism) for a Newtonian outflow model and two relativistic off-axis jet models. Here we assume the N. Aghanim et al. (2020) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The ratio of equipartition quantities derived from our analysis of AT2019dsg with that using the framework of T. Matsumoto & T. Piran (2023) for a Newtonian case (black) and off-axis case (colored) for no corrections C = 3, ξ = 1, p = 2, ϵ = 1 (left), for only hot proton and electrons radiating at νm corrections C ̸= 3, ξ ̸= 1, p ̸= 2, ϵ = 1 (middle), and for deviations from equipartition C ̸= 3, ξ ̸= 1, p… view at source ↗
Figure 4
Figure 4. Figure 4: The ratio between the equipartition quantities derived from our analysis of AT2019dsg and that of Y. Cendes et al. (2021), which uses the R. Barniol Duran et al. (2013) formalism, for a Newtonian model. Here we assume the same geometry as Y. Cendes et al. (2021) fV = 0.36, fA = 1, and same microphysical parameters ϵe = 0.1, ϵB = 0.02 and the N. Aghanim et al. (2020) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The probability density of ϵB assuming that the M. A. Keim et al. (2019) estimate for the radius of the emitting region R = (2.843 ± 0.302) × 1019 cm of J0231-0433 is normally distributed. The possible values of ϵB for an in equipartition ϵ = 1 scenario are for 2 < p < 3. There is a maximum ∼ 0.3% chance the system is in equipartition when considering a DFE ϵ ̸= 1 scenario [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 6
Figure 6. Figure 6: The equipartition radius and energy for the Newtonian, on axis, and off-axis cases for AT2019dsg including all correc￾tions C ̸= 3, ξ ̸= 1, p ̸= 2, ϵ ̸= 1. Here we assume the same cosmology as T. Matsumoto & T. Piran (2023) H0 = 69.6 km/s/Mpc, Ωm,0 = 0.286, Ω0 = 0.714 and the same geometric parameters fA = fV = 1. For the on axis solution we set γm = 2 as the γm we calculate from the bulk LF is small < 2 … view at source ↗

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Works this paper leans on

35 extracted references · 35 canonical work pages · 3 internal anchors

  1. [1]

    doi:10.1051/0004-6361/201833910 , abstract =

    Aghanim, N., Akrami, Y., Ashdown, M., et al. 2020, Astronomy & Astrophysics, 641, A6, doi: 10.1051/0004-6361/201833910

    work page doi:10.1051/0004-6361/201833910 2020

  2. [2]

    D., Berger, E., Guillochon, J., Zauderer, B

    Alexander, K. D., Berger, E., Guillochon, J., Zauderer, B. A., & Williams, P. K. G. 2016, ApJL, 819, L25, doi: 10.3847/2041-8205/819/2/L25

    work page doi:10.3847/2041-8205/819/2/l25 2016

  3. [3]

    M., Pavlovi´ c, M

    Arbutina, B., Uroˇ sevi´ c, D., Andjeli´ c, M. M., Pavlovi´ c, M. Z., & Vukoti´ c, B. 2012, ApJ, 746, 79, doi: 10.1088/0004-637X/746/1/79 Barniol Duran, R., Nakar, E., & Piran, T. 2013, The Astrophysical Journal, 772, 78, doi: 10.1088/0004-637X/772/1/78

    work page doi:10.1088/0004-637x/746/1/79 2012

  4. [4]

    doi:10.1002/asna.200510366 , keywords =

    Beck, R., & Krause, M. 2005, Astronomische Nachrichten, 326, 414, doi: 10.1002/asna.200510366

    work page doi:10.1002/asna.200510366 2005

  5. [5]

    Bell, A. R. 1978, MNRAS, 182, 443, doi: 10.1093/mnras/182.3.443

    work page doi:10.1093/mnras/182.3.443 1978

  6. [6]

    D., & Ostriker, J

    Blandford, R. D., & Ostriker, J. P. 1978, ApJL, 221, L29, doi: 10.1086/182658

    work page doi:10.1086/182658 1978

  7. [7]

    Brent, R. P. 1972, Algorithms for minimization without derivatives (Prentice-Hall)

    work page 1972

  8. [8]

    S., Carotenuto, F., Fender, R., et al

    Bright, J. S., Carotenuto, F., Fender, R., et al. 2025, The Astrophysical Journal, 981, 48, doi: 10.3847/1538-4357/adaaef

    work page doi:10.3847/1538-4357/adaaef 2025

  9. [9]

    Burbidge, G. R. 1956, ApJ, 124, 416, doi: 10.1086/146237

    work page doi:10.1086/146237 1956

  10. [10]

    L., & Barthel, P

    Carilli, C. L., & Barthel, P. D. 1996, A&A Rv, 7, 1, doi: 10.1007/s001590050001

    work page doi:10.1007/s001590050001 1996

  11. [11]

    D., Berger, E., et al

    Cendes, Y., Alexander, K. D., Berger, E., et al. 2021, The Astrophysical Journal, 919, 127, doi: 10.3847/1538-4357/ac110a

    work page doi:10.3847/1538-4357/ac110a 2021

  12. [12]

    Ubiquitous Late Radio Emission from Tidal Disruption Events

    Cendes, Y., Berger, E., Alexander, K. D., et al. 2023, Ubiquitous Late Radio Emission from Tidal Disruption Events, doi: 10.48550/arXiv.2308.13595

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2308.13595 2023

  13. [13]

    Chevalier, R. A. 1998, ApJ, 499, 810, doi: 10.1086/305676

    work page doi:10.1086/305676 1998

  14. [14]

    T., Alexander, K

    Christy, C. T., Alexander, K. D., Margutti, R., et al. 2024, The Astrophysical Journal, 974, 18, doi: 10.3847/1538-4357/ad675b

    work page doi:10.3847/1538-4357/ad675b 2024

  15. [15]

    Condon, J. J. 1992, ARA&A, 30, 575, doi: 10.1146/annurev.aa.30.090192.003043

    work page doi:10.1146/annurev.aa.30.090192.003043 1992

  16. [16]

    Towards improved synchrotron self absorption energy estimates: accounting for inhomogeneous and non-spherical emitting regions

    Cowie, F. J., & Fender, R. P. 2026, doi: 10.48550/arXiv.2606.11307

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.11307 2026

  17. [17]

    2019, MNRAS, 489, 4836, doi: 10.1093/mnras/stz2000

    Fender, R., & Bright, J. 2019, MNRAS, 489, 4836, doi: 10.1093/mnras/stz2000

    work page doi:10.1093/mnras/stz2000 2019

  18. [18]

    A., Waxman, E., & Kulkarni, S

    Frail, D. A., Waxman, E., & Kulkarni, S. R. 2000, ApJ, 537, 191, doi: 10.1086/309024

    work page doi:10.1086/309024 2000

  19. [19]

    L., & Syrovatskii, S

    Ginzburg, V. L., & Syrovatskii, S. I. 1965, ARA&A, 3, 297, doi: 10.1146/annurev.aa.03.090165.001501

    work page doi:10.1146/annurev.aa.03.090165.001501 1965

  20. [20]

    Gould, R. J. 1979, ˚ ap, 76, 306–311

    work page 1979

  21. [21]

    2004, International Journal of Modern Physics D, 13, 1549, doi: 10.1142/S0218271804005080

    Govoni, F., & Feretti, L. 2004, International Journal of Modern Physics D, 13, 1549, doi: 10.1142/S0218271804005080

    work page doi:10.1142/s0218271804005080 2004

  22. [22]

    A., Callingham, J

    Keim, M. A., Callingham, J. R., & R¨ ottgering, H. J. A. 2019, Astronomy & Astrophysics, 628, A56, doi: 10.1051/0004-6361/201936107

    work page doi:10.1051/0004-6361/201936107 2019

  23. [23]

    2021, The Astrophysical Journal Letters, 923, L14, doi: 10.3847/2041-8213/ac3d97

    Margalit, B., & Quataert, E. 2021, The Astrophysical Journal Letters, 923, L14, doi: 10.3847/2041-8213/ac3d97

    work page doi:10.3847/2041-8213/ac3d97 2021

  24. [24]

    2024, The Astrophysical Journal, 977, 134, doi: 10.3847/1538-4357/ad8b47

    Margalit, B., & Quataert, E. 2024, The Astrophysical Journal, 977, 134, doi: 10.3847/1538-4357/ad8b47

    work page doi:10.3847/1538-4357/ad8b47 2024

  25. [25]

    Marscher, A. P. 1983, The Astrophysical Journal, 264, 296, doi: 10.1086/160597

    work page doi:10.1086/160597 1983

  26. [26]

    2023, Monthly Notices of the Royal Astronomical Society, 522, 4565–4576, doi: 10.1093/mnras/stad1269

    Matsumoto, T., & Piran, T. 2023, Monthly Notices of the Royal Astronomical Society, 522, 4565–4576, doi: 10.1093/mnras/stad1269

    work page doi:10.1093/mnras/stad1269 2023

  27. [27]

    1980, ARA&A, 18, 165, doi: 10.1146/annurev.aa.18.090180.001121

    Miley, G. 1980, ARA&A, 18, 165, doi: 10.1146/annurev.aa.18.090180.001121

    work page doi:10.1146/annurev.aa.18.090180.001121 1980

  28. [28]

    Moffet, A. T. 1975, in Galaxies and the Universe, ed. A. Sandage, M. Sandage, & J. Kristian, 211

    work page 1975

  29. [29]

    Pacholczyk, A. G. 1970, Radio astrophysics. Nonthermal processes in galactic and extragalactic sources

    work page 1970

  30. [30]

    2026, equipartition, 1.0.0 Zenodo, doi: 10.5281/zenodo.20835476

    Rohde, C., Franz, N., Laskar, T., & Christy, C. 2026, equipartition, 1.0.0 Zenodo, doi: 10.5281/zenodo.20835476

    work page doi:10.5281/zenodo.20835476 2026

  31. [31]

    A., & Readhead, A

    Scott, M. A., & Readhead, A. C. S. 1977, Monthly Notices of the Royal Astronomical Society, 180, 539–550, doi: 10.1093/mnras/180.4.539

    work page doi:10.1093/mnras/180.4.539 1977

  32. [32]

    The First Radio-Bright Off-Nuclear TDE 2024tvd Reveals the Fastest-Evolving Double-Peaked Radio Emission

    Sfaradi, I., Margutti, R., Chornock, R., et al. 2025, The First Radio-Bright Off-Nuclear TDE 2024tvd Reveals the Fastest-Evolving Double-Peaked Radio Emission, https://arxiv.org/abs/2508.03807

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  33. [33]

    2009, title Clustering of luminous red galaxies - III

    Shen, R.-F., & Zhang, B. 2009, Monthly Notices of the Royal Astronomical Society, 398, 1936–1950, doi: 10.1111/j.1365-2966.2009.15212.x Uroˇ sevi´ c, D., Pavlovi´ c, M. Z., & Arbutina, B. 2018, ApJ, 855, 59, doi: 10.3847/1538-4357/aaac2d

    work page doi:10.1111/j.1365-2966.2009.15212.x 2009

  34. [34]

    Wijers, R. A. M. J., & Galama, T. J. 1999, The Astrophysical Journal, 523, 177, doi: 10.1086/307705

    work page doi:10.1086/307705 1999

  35. [35]

    and Berger, E

    Zauderer, B. A., Berger, E., Soderberg, A. M., et al. 2011, Nature, 476, 425, doi: 10.1038/nature10366

    work page doi:10.1038/nature10366 2011