Novel Scalings of Neutron Star Properties from Analyzing Dimensionless Tolman--Oppenheimer--Volkoff Equations

Bao-An Li; Bao-Jun Cai

REVIEW 4 major objections 5 minor 2 cited by

Novel Scalings of Neutron Star Properties from Analyzing Dimensionless Tolman--Oppenheimer--Volkoff Equations

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This review claims that neutron star core properties can be read directly from the dimensionless Tolman–Oppenheimer–Volkoff equations, without any nuclear equation-of-state model, through universal scalings capped by causality at…

desk verdict A useful review of the IPAD-TOV program, but the claimed EOS-free derivations rest on an uncontrolled truncation at the stellar surface, so the headline results are not secure. read the letter →

arxiv 2501.18676 v1 pith:JSIK62VL submitted 2025-01-30 astro-ph.HE astro-ph.SRgr-qcnucl-exnucl-th

Bao-Jun Cai , Bao-An Li This is my paper · ORCID

classification astro-ph.HEastro-ph.SRgr-qcnucl-exnucl-th

keywords neutronstarequationofstatedimensionlessTOVequationsperturbativeexpansionmass-radiusrelationcompactnessspeedsoundsquaredcausalityboundtraceanomaly

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 Tolman–Oppenheimer–Volkoff equations, when rewritten in dimensionless variables, can be solved perturbatively about the stellar center without ever specifying a nuclear equation of state. The central claim is that the reduced pressure and energy density admit polynomial expansions whose coefficients are fixed by the TOV equations alone, leading to universal scalings of neutron star radius, mass, and compactness with combinations of the central pressure ratio $X=P_c/\varepsilon_c$, the central energy density, and the central speed of sound. If the claim holds, a measured mass, radius, or compactness directly fixes the central pressure, energy density, and stiffness of the densest visible matter, and causality sharpens the allowed central pressure ratio from $X\le 1$ to $X\lesssim 0.374$. The same analysis is used to exclude a constant-speed-of-sound linear EOS near neutron-star centers and to attribute the peak in the sound-speed profile of massive neutron stars to strong-field gravity rather than to the nuclear EOS.

What carries the argument

The central machinery is the IPAD-TOV approach: recast the TOV equations in dimensionless variables $\hat r=r/Q$, $\hat P=P/\varepsilon_c$, $\hat\varepsilon=\varepsilon/\varepsilon_c$ with $Q=(4\pi\varepsilon_c)^{-1/2}$, then solve them perturbatively near the center using the double expansion in the small quantities $X=\hat P_c$ and $\mu=\hat\varepsilon-1$. The load-bearing object is the expansion of the reduced pressure, with coefficients $b_2=-(1+3X^2+4X)/6$ and higher $b_k$; the radius is obtained from the zero of the truncated pressure, yielding $\hat R^2=6X/(1+3X^2+4X)$, and the mass and compactness scalings follow by integration. The same expansion produces the central sound-speed formula and the conditions under which a peaked sound-speed profile appears, tied to the sign of $a_4$ and the GR factor in the TOV pressure equation.

What would settle it

Take any of the paper's $10^5$ meta-model EOSs, solve the full TOV equations, and compare the exact surface radius and maximum mass with $\hat R^2=6X/(1+3X^2+4X)$ and the associated scalings; if the residuals at $\hat R\approx 1$ are not consistent with the stated regression accuracy of about 0.3 km and 0.002 $M_\odot$, the central claim fails.

Watch

Extended reading notes

Core claim

The paper's discovery is that the dimensionless TOV equations contain the central EOS implicitly: writing $\hat P(\hat r)=X+b_2\hat r^2+b_4\hat r^4+\cdots$ and $\hat\varepsilon(\hat r)=1+a_2\hat r^2+a_4\hat r^4+\cdots$, the coefficients $b_2=-(1+3X^2+4X)/6$, $b_4$, and their counterparts are fixed by the equations themselves, with $a_2=b_2/s_c^2$. Setting $\hat P(\hat R)=0$ at the truncated order gives $\hat R^2=6X/(1+3X^2+4X)$, from which follow the $\nu_c$, $\Gamma_c$, and $\Pi_c$ scalings for radius, mass, and compactness. The authors show numerically with a large ensemble of EOSs that these scalings hold for both the maximum-mass TOV configuration and for general stable stars, and they derive the central speed-of-sound formula $s_c^2=X\left[1+\frac{1+\Psi}{3}\frac{1+3X^2+4X}{1-3X^2}\right]$ whose causality bound $s_c^2\le 1$ gives $X\lesssim 0.374$. They also derive a lower bound on the trace anomaly and argue that the peaked sound-speed profile in massive neutron stars is caused by the GR correction factor $(1-2\hat M/\hat r)^{-1}$ rather than by any specific EOS.

Load-bearing premise

The load-bearing premise is that the power-series expansion of pressure and energy density around the center, cut off after the first correction term, is accurate all the way to the neutron-star surface; the paper does not prove that the omitted fourth-order terms are negligible at $\hat R\approx 1$.

Editorial extensions

If this is right

Observed neutron-star masses, radii, and compactnesses can be converted directly into central pressure, central energy density, and central speed of sound, bypassing nuclear-EOS modeling.
A constant-speed-of-sound linear EOS of the form $P=\zeta\varepsilon+\Phi$ is excluded near neutron-star centers by the singularity structure of the dimensionless TOV equations.
Causality tightens the central pressure ratio to $X\lesssim 0.374$, implying a compactness bound $\xi\lesssim 0.264$ and a lower bound on the trace anomaly for supra-dense matter.
The maximum mass of stable neutron stars is estimated as about $2.26$–$2.28\,M_\odot$ with radius near $12$ km, independent of EOS-model input.
A peak in the density profile of the speed of sound in massive neutron stars is a geometric, strong-field gravity effect rather than a nuclear-EOS effect.

Reading between the lines

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

The scalings are established by numerical verification rather than by a proof that the truncated expansion converges at the stellar surface; if the omitted $b_4\hat R^4$ terms are not negligible at $\hat R\approx 1$, the numerical coefficients in the radius scaling would shift and the extracted values of $X$ would change.
If the causality bound $X\lesssim 0.374$ survives sharper observational constraints, it provides a model-independent ceiling that any future EOS construction, hadronic, hybrid, or quarkyonic, must obey at neutron-star central densities.
The same dimensionless expansion could be applied to tidal deformability or moment of inertia, potentially producing EOS-independent relations between those observables and the central ratio $X$; the paper does not carry out that extension.
Because the TOV equations are blind to composition, even a successful extraction of $P_c(\varepsilon_c)$ leaves the particle content of the core unconstrained; linking $P_c(\varepsilon_c)$ to nuclear symmetry energy still requires a microscopic model.

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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

Honestly, this is a review of your own earlier work rather than a new research paper. That's fine as a genre, and the paper does a good job of laying out the IPAD-TOV approach, its scalings, and the many applications. The numerical evidence that the scaled TOV equations generate quasi-universal relations between Rmax and νc, Mmax and Γc, and ξ and Πc is real: 104 phenomenological and 10^5 meta-model EOSs all fall on tight lines. That is a useful empirical fact, and the paper deserves credit for documenting it carefully.

The problem is that the paper treats these relations as derived from the TOV equations without a model, when the derivation actually uses a power-series expansion in rhat truncated at order rhat^2 and then evaluates it at the stellar surface, rhat ~ 1. The stress-test is right: the b4 term is comparable to X at the surface, so the truncation is not controlled there. The paper's own Fig. 24 limits the expansion to rhat ≲ 0.2. Consequently, the radius scaling (Eq. 4.4), the mass scaling, the central speed-of-sound formula, and the causal bound X≤0.374 all inherit this uncontrolled step. The later linear fits (Eqs. 4.30-4.31, 4.66-4.68) are calibrated to EOS ensembles, so "EOS-model independent" is an overstatement. This is a load-bearing issue for the paper's central claim, not a minor slip. The authors themselves hedge in the introduction, saying the scalings are "largely instead of absolutely independent" of nuclear EOS models, but the headline claims do not consistently respect that qualification.

The exclusion of a linear EOS is also shaky: the singular solution in Eq. (2.7) doesn't satisfy the regular central boundary conditions Phat(0)=X, Mhat(0)=0, and standard CSS models do produce regular stars. The discussion about the generalized linear EOS has the same flavor. The paper reviews many other topics—symmetry energy, peaked speed of sound, trace anomaly—and those sections are informative, though they also depend on the same perturbative framework.

Who should read this? People who want a one-stop summary of the IPAD-TOV program and its empirical scaling relations. The scalings might be a useful heuristic for quick estimates, but I wouldn't rely on the derivation as a proof. The paper deserves a serious referee, but the referee should ask the authors to either prove that the truncation error is small at the surface or clearly reposition the scalings as empirical fits, not EOS-free derivations. As it stands, the central claims overreach the math.

Referee Report

4 major / 5 minor

Summary. The manuscript reviews the authors' IPAD-TOV program, in which the TOV equations are recast in dimensionless form and solved perturbatively near the stellar center by expanding the reduced pressure, energy density and mass in powers of the reduced radial coordinate. From the leading-order terms it derives scaling relations connecting NS radius, mass and compactness to combinations of the central pressure-to-energy-density ratio X=Pc/εc and central density εc (Eqs. 4.6-4.8); claims to extract the central EOS of PSR J0740+6620, PSR J0030+0451, PSR J0437-4715 and canonical NSs directly from observations without using nuclear EOS models; argues that linear or constant-speed-of-sound EOSs are excluded near NS centers; derives a causality bound X≤0.374; explains the near-vertical shapes of M-R curves; attributes a peaked speed-of-sound profile to strong-field gravity; and estimates MTOV≈2.26±0.28 Msun with RTOV≈12.62±1.51 km. The scalings are tested against 104 tabulated EOSs and 10^5 randomly generated meta-model EOSs, with reported Pearson coefficients and standard errors. The article is largely a review of the authors' earlier works in Refs. [108,112-115].

Significance. If the analytical derivation were controlled, the paper would represent a significant advance: EOS-model-independent extraction of the central EOS from mass-radius data, a universal compactness-central-pressure relation, and falsifiable bounds on central pressure and maximum mass. The empirical verification is a genuine strength; the ξ-Πc (r≈0.972), MNS-Γc (r≈0.965) and Rmax-νc correlations, with the reported small standard errors, are useful candidate universal relations, and the comparison with the Lattimer-Prakash empirical radius scaling is informative. The survey of observational and theoretical constraints in Section 3 is thorough and well documented. The significance is nonetheless conditional, because the two load-bearing analytical claims (the exclusion of CSS EOSs and the evaluation of the near-center expansion at the stellar surface) are not supported by the derivations as written, and the quoted extraction errors do not include the systematic truncation uncertainty.

major comments (4)

[2.1, Eqs. (2.7)-(2.15)] The exclusion of the linear EOS P=ζε and its constant-speed-of-sound generalization is not established. The exact solution (2.7) has Phat and εhat diverging as rhat→0, so it does not satisfy the physical central boundary conditions Phat(0)=ζ, εhat(0)=1; it is a singular branch, not the star obtained by integrating the TOV equations from a finite central pressure. For the same EOS, the paper's own regular near-center expansion of Section 2.3 (Eqs. (2.22)-(2.24) with X=ζ, together with a2=b2/ζ from Eq. (2.34)) provides a perfectly regular local solution, so the statement that a linear EOS is fundamentally inconsistent with the nature of the TOV equations (text after Eq. (2.7)) is a non-sequitur and is internally contradicted. The M-R relation (2.9)-(2.10) derived from the singular branch therefore cannot be used as evidence against CSS EOSs in NS cores.
[4.1, Eq. (4.4)] The radius scaling (4.4), Rhat=[6X/(1+3X^2+4X)]^{1/2}, is obtained by imposing Phat(Rhat)≈X+b2 Rhat^2=0, i.e., by applying the near-center quadratic expansion at the stellar surface. The expansion is controlled only near the center: the paper's own Fig. 24 and the discussion in Section 2.3 limit its reliability to rhat≲0.2, about one-fifth of the reduced radius, while Eq. (2.26) estimates Rhat≈1.1 for a 2 Msun NS. For a typical X≈0.24, Eqs. (2.30) and (2.35) give b2≈-0.356 and b4≈0.17-0.21; at Rhat≈1 the quartic term b4 Rhat^4 is of the same order as |X|, and the truncated quartic X+b2 rhat^2+b4 rhat^4 has no real root. Hence the mass scaling (4.7), compactness scaling (4.8), central SSS (4.15)/(4.17), the bound (4.18) and the central-EOS extractions in Tables 2-3 are not controlled consequences of the TOV equations, and no truncation error estimate is provided for evaluating the series at Rhat≈1.
[4.2-4.5, Eqs. (4.30)-(4.31), (4.66)-(4.68), Tables 2-3] The claim that the approach extracts the central EOS without using any nuclear EOS model (Abstract, Fig. 1) is not supported by the manuscript as written. The dimensionless-TOV analysis supplies the functional forms of the scalings, but all numerical coefficients used in the extraction (AM≈1242, BM≈-0.08, AR≈572, BR≈4.22 in Eqs. (4.66)-(4.68), and Amax_M, Bmax_M, Amax_R, Bmax_R in Eqs. (4.30)-(4.31)) are linear-regression fits to 104 or 10^5 model EOSs. The paper concedes this in Section 1 (largely instead of absolutely independent), which is in tension with the abstract. Moreover, the quoted central values and errors (e.g., εc=901(+214,-287) MeV/fm^3 and Pc=218(+93,-125) MeV/fm^3 for PSR J0740+6620 in Table 2) propagate only the observational and fit uncertainties; the systematic error from the surface truncation of Eq. (4.4) is not included, so the precision implied by these numbers is not justified.
[6.1-6.9, Eqs. (4.17)-(4.18), (4.95)] The causality bound X≤0.374 is advertised as a fundamental GR+causality limit, but it is obtained from s2c≤1 using Eq. (4.17), which is itself built on the truncated mass scaling; the bound therefore inherits the uncontrolled surface truncation discussed above. Its subsequent use is further weakened by a partial circularity: the scaling coefficients in Eqs. (4.67) and (4.96) are fitted to the same EOS ensembles from which the bound is then used to exclude EOSs (Fig. 16; Section 4.8) and to derive MTOV≈2.26±0.28 Msun and RTOV≈12.62±1.51 km (Eqs. (4.90)-(4.95)). The manuscript should either prove the bound directly from the untruncated TOV equations with a quantitative error estimate, or present it as an empirical property of the EOS ensemble used.

minor comments (5)

[2.3, Eq. (2.48) and Fig. 22 caption] Equation (2.48) and the caption of FIG. 22 contain corrupted or garbled characters (e.g., the trailing token in Eq. (2.48) and the expression CΦ δ2 ˆr + O(δ4 ˆr) in the caption); these should be repaired.
[Notation, Eqs. (5.15), (4.59)] The symbol ε is used with two different normalizations: ε≡ε/ε0 in Eq. (5.15) and Table 1, while the double expansion (4.59) is formulated in µ=εhat-1 with εhat=ε/εc; the proximity of these notations invites confusion, and a short summary of the three normalizations (ε, εhat, Y) used in Sections 4-6 would help the reader.
[Table 2] Table 2 reports s2c for five different radius inputs but does not state whether Eq. (4.15) with Ψ=0 or a finite-Ψ variant was used; adding the formula and the value of Ψ used would make the table reproducible.
[Abstract and Section 1] The abstract states the exclusions and bounds as established results, while Section 1 concedes that the scalings are largely instead of absolutely independent of EOS models and Section 4.8 notes that the radius scaling is obtained by truncation; the abstract should be made consistent with these caveats.
[4.5, Eqs. (4.70)-(4.71)] The next-to-leading-order correction κ1=18/25 to the mass scaling is stated without derivation; since the leading-order scalings are the paper's main tool, the reader needs to know whether this coefficient also results from evaluating a truncated series at the surface or from the recurrence relations of Section 2.3.

Circularity Check

2 steps flagged · score 5.0 of 10

Central EOS 'extraction without nuclear EOS models' is performed by inverting scaling fits made on 10^4-10^5 EOS ensembles; the TOV-derived functional form is real, but the numerical predictions inherit the fitted calibration.

fitted input called prediction [Sec. 4.2, Eqs. (4.30)-(4.31), used in Eqs. (4.33)-(4.34) and Table 2]
"By performing linear fits of the results obtained from using these EOS samples in solving the TOV equations in the traditional approach, the quantified scaling relations are determined to be R_max/km≈A_R ν_c+B_R≈1.05×10^3(ν_c/[fm^{3/2}/MeV^{1/2}])+0.64 ... and M_max/M_⊙≈A_M Γ_c+B_M≈1.73×10^3(Γ_c/[fm^{3/2}/MeV^{1/2}])−0.106."

The paper's signature claim is that IPAD-TOV extracts the central EOS 'without using any nuclear EOS model.' But the practical extraction uses the linear fits (4.30)-(4.31), whose slope and intercept were obtained by least-squares fitting 104 phenomenological/microscopic EOSs solved with the traditional TOV approach. Equation (4.34) then defines P_c(ε_c) via f_R=(R_max−0.64)/1050, so the fitted intercept/slope enter the extracted ε_c, P_c and s_c^2 directly (Table 2). Thus the 'model-independent' central EOS is, by construction of the inversion formula, a projection of the fitted EOS ensemble; the dimensionless TOV equations fix the functional form, but the numerical extraction is calibrated on the ensemble. This is fitted input called prediction.
fitted input called prediction [Sec. 4.5, Eqs. (4.66)-(4.68), used in Eq. (4.72) and Table 3]
"To test the mass, radius and compact scalings of Eqs. (4.7), (4.6) and (4.8), we randomly generate 10^5 meta-model EOSs ... Quantitatively, we have ξ≈A_ξ Π_c+B_ξ≈2.31 Π_c−0.032, M_NS/M_⊙≈A_M Γ_c+B_M≈1242 Γ_c−0.08, R/km≈A_R ν_c+B_R≈572 ν_c+4.22."

These 10^5-EOS fits are then inverted to give ε_c=Π_c^3 (A_M/(M_NS/M_⊙−B_M))^2 and P_c=X ε_c for PSR J0740+6620, PSR J0030+0451 and the canonical NS (Eq. 4.72 and Table 3). The 'mass+compactness' extraction is therefore the inverse of the fitted scaling: the resulting central EOS values are what the meta-model ensemble predicts, not information obtained from the dimensionless TOV equations alone. The paper itself states that the scalings are 'largely instead of absolutely independent of nuclear EOS models,' yet the central extraction is presented without that caveat, making the claimed EOS-model-free prediction statistically forced by the fitted ensemble.

full rationale

The functional scalings R∼ν_c, M∼Γ_c and ξ∼Π_c genuinely follow from the dimensionless TOV equations via the perturbative expansion, and the exclusion of a constant-speed-of-sound linear EOS is a legitimate analytic result, not a circular step. The main circularity concern is confined to the quantitative extraction of central EOS values: the constants used to invert observables into ε_c, P_c and s_c^2 are least-squares fits to 104 and 10^5 model EOSs, so the repeated claim that the central EOS is obtained 'without using any nuclear EOS model' is not supported by the derivation. The uncontrolled truncation at r^4 in the radius scaling is a correctness and convergence risk, not a circularity, and I do not count it here. There is no evidence that the central arguments rest on unverified self-citations; the derivations are displayed in the paper. Overall, the central claim is partially circular because the numerical 'predictions' reduce to the fitted calibration relations, even though the scaling forms themselves have independent TOV-based content.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The central extraction scheme depends on fitted scaling coefficients (AM, BM, AR, BR, Aξ, Bξ, plus the TOV-configuration fits) calibrated on model EOS ensembles, so the ledger records them as free parameters. It also relies on several unproved domain assumptions: convergence of the central expansion out to the surface, representativeness of the meta-model ensemble, and applicability of causality. No new physical entities are introduced.

free parameters (9)

AM (mass scaling coefficient) = 1242 ± 15 fm^{3/2}/MeV^{1/2}
Linear fit to 10^5 meta-model EOSs, Eq. (4.67), used to convert observed mass to central EOS.
BM (mass scaling intercept) = -0.08 ± 0.02 M_sun
Linear fit to 10^5 meta-model EOSs, Eq. (4.67), used in mass scaling.
AR (radius scaling coefficient) = 572 ± 25 fm^{3/2}/MeV^{1/2}
Linear fit to 10^5 meta-model EOSs, Eq. (4.68), used to convert observed radius to central EOS.
BR (radius scaling intercept) = 4.22 ± 0.35 km
Linear fit to 10^5 meta-model EOSs, Eq. (4.68).
Aξ (compactness scaling coefficient) = 2.31 ± 0.03
Linear fit to 10^5 meta-model EOSs, Eq. (4.66), used to extract X from compactness.
Bξ (compactness scaling intercept) = -0.032 ± 0.003
Linear fit to 10^5 meta-model EOSs, Eq. (4.66).
A_M^max, B_M^max (TOV mass scaling) = 1730 ± 30 fm^{3/2}/MeV^{1/2}, -0.106 ± 0.035 M_sun
Linear fit to 104 EOS samples, Eq. (4.31), used for TOV-configuration mass scalings.
A_R^max, B_R^max (TOV radius scaling) = 1050 ± 30 fm^{3/2}/MeV^{1/2}, 0.64 ± 0.25 km
Linear fit to 104 EOS samples, Eq. (4.30).
aΨ, bΨ (Ψ-MNS relation) = -1.62 ± 0.13, 5.12 ± 0.22
Fitted to meta-model EOSs, Eq. (4.73), used for vertical M-R shape analysis.

assumptions (5)

domain assumption The dimensionless TOV equations are solved by polynomial expansions around the center and the truncation at rhat^2 is valid up to the surface.
Used to obtain Rhat from Phat(Rhat)≈0 and all scaling relations; see Eq. (4.4).
ad hoc to paper The 10^5 meta-model EOS ensemble spans all plausible NS EOSs.
Used to fit scaling coefficients and claim EOS-model independence; see Section 4.5.
domain assumption NS matter near the center admits a convergent double expansion in X and µ with coefficients O(1).
Needed for perturbative expansions in Sections 2-4.
domain assumption The strongest observed massive NS can be approximated as being at the TOV configuration.
Used to extract central EOS of PSR J0740+6620 via Mmax scalings; see Section 4.2.
standard math Causality s^2≤1 is applicable to NS matter.
Used to bound X≤0.374 and ξ; standard assumption about EOS causality.

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Cite this review

Pith. "Pith review of Novel Scalings of Neutron Star Properties from Analyzing Dimensionless Tolman--Oppenheimer--Volkoff Equations." pith.science (2026). https://pith.science/paper/JSIK62VL

@misc{pith2026250118676,
  author       = {Pith},
  title        = {Pith review of: Novel Scalings of Neutron Star Properties from Analyzing Dimensionless Tolman--Oppenheimer--Volkoff Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSIK62VL}},
  note         = {Machine review of arXiv:2501.18676}
}

read the original abstract

The TOV equations govern the radial evolution of pressure and energy density in static neutron stars (NSs) in hydrodynamical equilibrium. Using the reduced pressure and energy density with respect to the NS central energy density, the original TOV equations can be recast into dimensionless forms. While the traditionally used integral approach for solving the original TOV equations require an input nuclear Equation of State (EOS), the dimensionless TOV equations can be anatomized by using the reduced pressure and energy density as polynomials of the reduced radial coordinate without using any input nuclear EOS. Interesting and novel perspectives about NS core EOS can be extracted directly from NS observables using this new approach based on Intrinsic and Perturbative Analyses of the Dimensionless (IPAD) TOV equations (IPAD-TOV). In this review, we first discuss the length and energy density scales of NSs as well as the dimensionless TOV equations for scaled variables and their perturbative solutions near NS cores. We then review several new insights into NS physics gained from using the IPAD-TOV. We also demonstrate that the strong-field gravity plays a fundamental role in extruding a peak in the density/radius profile of the speed of sound squared (SSS) in massive NS cores independent of the nuclear EOS. Finally, some future perspectives of NS research using the IPAD-TOV are outlined.

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