arxiv: 2604.09764 · v1 · submitted 2026-04-10 · 🌌 astro-ph.SR · astro-ph.HE

Eccentricities of millisecond pulsars with intermediate-mass progenitors

Hagai Bareli , Sivan Ginzburg This is my paper

Pith reviewed 2026-05-10 17:34 UTC · model grok-4.3

classification 🌌 astro-ph.SR astro-ph.HE

keywords millisecond pulsarsCO white dwarfsRoche-lobe overflowbinary evolutioneccentricityconvective envelopemass transfer

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The pith

Intermediate-mass stars form millisecond pulsars with CO white dwarf companions whose eccentricities match those from lower-mass progenitors because eccentricity scales only weakly with envelope mass at detachment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models the stable Roche-lobe overflow channel that produces millisecond pulsars with CO white dwarf companions from intermediate-mass stars. It supplies a simple analytical relation between the final orbital period and the white dwarf mass together with the progenitor initial mass. The eccentricity is calculated for the first time from the fluctuating gravitational quadrupole moment of the convective envelope at the instant of detachment. Because eccentricity depends only on the one-sixth power of envelope mass, the higher envelope mass at detachment for intermediate-mass stars barely changes the resulting eccentricity, explaining the observed similarity to systems with helium white dwarfs.

Core claim

The eccentricity e in this process is set by the fluctuating gravitational quadrupole moment of the progenitor's convective envelope during Roche-lobe detachment. Intermediate-mass progenitors detach when their non-degenerate cores ignite helium, in contrast to low-mass stars that detach when their envelopes become too light to support a burning shell. Despite the order of magnitude higher envelope mass at detachment m_e, the eccentricity is barely affected because e ∝ m_e^{1/6}, explaining why intermediate-mass CO white dwarfs have similar eccentricities to lower mass helium white dwarfs. Massive CO and ONe white dwarfs probably formed through a different channel of unstable Roche-lobe flow

What carries the argument

The relation e ∝ m_e^{1/6} that follows from the fluctuating gravitational quadrupole moment of the convective envelope at Roche-lobe detachment, combined with the timing of detachment at helium ignition in the non-degenerate core.

If this is right

The final orbital period follows an analytical function of progenitor initial mass and white dwarf mass.
CO white dwarfs with masses up to about 0.6 solar masses arise from this stable-transfer channel and inherit the same eccentricity distribution as helium white dwarfs.
Massive CO and ONe white dwarfs form instead through unstable mass transfer during helium shell burning followed by common-envelope inspiral.
The eccentricities measured for those massive white dwarfs must be produced by the common-envelope channel.

Where Pith is reading between the lines

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

The same quadrupole-fluctuation mechanism could be applied to other binaries that detach while still possessing thick convective envelopes.
A statistical comparison of orbital periods versus white dwarf masses in observed systems would directly test the analytical period-mass relation.
Future eccentricity measurements in the massive white dwarf population could cleanly separate the two formation channels.

Load-bearing premise

Eccentricity is produced only by fluctuations in the gravitational quadrupole moment of the convective envelope at the moment of Roche-lobe detachment, and intermediate-mass stars detach precisely when their non-degenerate cores ignite helium.

What would settle it

A sample of millisecond pulsars with CO white dwarf companions that shows eccentricities deviating by more than a factor of a few from the m_e^{1/6} prediction, or a clear absence of the predicted relation between orbital period and white dwarf mass in a large population.

Figures

Figures reproduced from arXiv: 2604.09764 by Hagai Bareli, Sivan Ginzburg.

**Figure 1.** Figure 1: Evolutionary tracks of low-mass (solid grey lines) and intermediatemass (solid blue lines) single stars, with the stellar mass /M⊙ indicated near each track. As stars exit the main sequence, they develop a He core with a growing mass c while their envelope’s radius expands. Low-mass stars harbour degenerate cores and expand on almost the same ∝ 9/2 c curve (dashed red line), given by equation (1). Interme… view at source ↗

**Figure 2.** Figure 2: Observed MSP–He WD (grey squares) and MSP– CO WD (blue circles, these include ONe WDs) field binaries with pulsar spin periods below 50 ms from the ATNF Pulsar Catalogue http://www.atnf.csiro.au/research/pulsar/psrcat (Manchester et al. 2005), version 2.7.0 (October 2025). Markers represent median WD masses, and the error bars span from the minimum mass to the 90th percentile. Low-mass He WDs are explaine… view at source ↗

**Figure 3.** Figure 3: The parameter space (initial donor mass , initial neutron star mass ns, and initial orbital period ) for stable Roche-lobe overflow of intermediate-mass progenitors (i.e. the channel we focus on in this study). Due to the donor to accretor mass ratio, low-mass progenitors ( ® 2 M⊙) are always stable, whereas massive progenitors ( ¦ 6 M⊙) are always unstable. For intermediate masses, the stability depends o… view at source ↗

**Figure 4.** Figure 4: The remaining envelope mass e (as a function of the initial progenitor mass and the final WD mass wd) shortly after Roche-lobe detachment, when the envelope’s radius contracts by a factor of 2 compared to the Roche lobe – freezing the eccentricity (we are not sensitive to the exact factor because of the sharp contraction; see Cohen et al. 2024). The intermediate-mass progenitors presented here (3 M⊙ ® ® 5 … view at source ↗

read the original abstract

One channel to form millisecond pulsars with CO white dwarf companions is through the stable Roche-lobe overflow of intermediate-mass ($3\,{\rm M}_\odot\lesssim M\lesssim 5\,{\rm M}_\odot$) stars at the end of the main sequence (Case A) or the beginning of the hydrogen shell burning phase (Case B). We reproduce previous numerical calculations of this channel and supplement them with a simple analytical model that relates the final orbital period $P(M,m_{\rm wd})$ to the white dwarf's mass and to its progenitor's initial mass $M$. We also theoretically calculate for the first time the eccentricity $e$ in this process, which is set by the fluctuating gravitational quadrupole moment of the progenitor's convective envelope during Roche-lobe detachment. Intermediate-mass progenitors detach when their non-degenerate cores ignite helium, in contrast to low-mass ($M\lesssim 2\,{\rm M}_\odot$) stars with degenerate cores that detach when their envelopes become too light to support a burning shell. Despite the order of magnitude higher envelope mass at detachment $m_{\rm e}$ in our case, the eccentricity is barely affected because $e\propto m_{\rm e}^{1/6}$, explaining why intermediate-mass ($m_{\rm wd}\lesssim 0.6\,{\rm M}_\odot)$ CO white dwarfs have similar eccentricities to lower mass helium white dwarfs. Massive CO and ONe white dwarfs ($m_{\rm wd}\gtrsim 0.6\,{\rm M}_\odot)$, on the other hand, probably formed through a different channel of unstable Roche-lobe overflow during helium shell burning (Case C), followed by common envelope inspiral. The measured eccentricities of these massive white dwarfs remain to be explained.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper provides a useful analytical model and the first eccentricity calculation for the intermediate-mass millisecond pulsar channel, with the main caveat being the assumed detachment timing.

read the letter

The main takeaway from this paper is its first analytical calculation of the eccentricity for millisecond pulsars in the intermediate-mass progenitor channel. It shows that the eccentricity is set by the fluctuating gravitational quadrupole of the convective envelope at detachment, leading to only weak dependence on the envelope mass and thus similar values to the low-mass helium white dwarf cases. The authors do well by reproducing previous numerical calculations with a simple analytical model relating the final orbital period to the white dwarf mass and the progenitor's initial mass. This makes the results more transparent and easier to use for estimates. The derivation of the eccentricity scaling, e proportional to m_e to the power of one sixth, comes directly from the physical mechanism rather than any fitting, which strengthens the claim. The soft spot is the key assumption about detachment timing. The work states that intermediate-mass stars detach when their non-degenerate cores ignite helium, resulting in higher envelope mass compared to degenerate cases. This premise drives the m_e value and the explanation for eccentricity similarity. The abstract does not detail extensive tests against full binary evolution grids with varying parameters like mass transfer efficiency or metallicity, so the robustness of that specific epoch could be probed further in revisions. If the timing shifts, the quantitative predictions change even if the scaling holds. This paper is aimed at specialists in stellar binary evolution and the formation of compact objects like pulsars. Readers interested in analytical models for binary outcomes or in understanding eccentricity distributions in pulsar-white dwarf systems will find it valuable. It is coherent and engages honestly with the literature on these channels, so it deserves a serious referee. I recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the formation of millisecond pulsars with CO white dwarf companions from intermediate-mass progenitors (3-5 M⊙) via stable Roche-lobe overflow in Case A or early Case B. It reproduces prior numerical binary evolution calculations, introduces a simple analytical model relating the final orbital period P to the white dwarf mass m_wd and progenitor initial mass M, and provides the first theoretical calculation of the eccentricity e. This eccentricity is attributed to the fluctuating gravitational quadrupole moment of the convective envelope at Roche-lobe detachment, which occurs when non-degenerate cores ignite helium (in contrast to low-mass degenerate-core systems). The key result is that e scales only weakly as m_e^{1/6} despite an order-of-magnitude larger envelope mass m_e, explaining the similarity in observed eccentricities between these CO white dwarfs (m_wd ≲ 0.6 M⊙) and lower-mass helium white dwarfs. More massive CO/ONe white dwarfs are suggested to form via unstable Case C mass transfer and common-envelope evolution.

Significance. If the central results hold, the work provides a useful analytical framework for orbital periods and a physical mechanism for eccentricity generation in these binaries, potentially unifying observations across white dwarf types via the weak m_e dependence. Reproducing prior numerical calculations adds credibility, and the first-principles eccentricity derivation is a clear strength that could inform population synthesis and channel discrimination. The insight that eccentricity is barely affected by higher envelope mass is noteworthy for explaining data on millisecond pulsar binaries.

major comments (2)

[Section describing the evolutionary channel and detachment timing] The central claim that intermediate-mass systems detach at non-degenerate helium ignition (setting m_e and enabling the e ∝ m_e^{1/6} explanation) is load-bearing but rests on an assumption whose generality is not demonstrated. No comparisons are shown to full binary-evolution grids varying initial mass, metallicity, or mass-transfer efficiency to confirm the detachment epoch holds across the 3-5 M⊙ range.
[Section on eccentricity derivation] The eccentricity calculation (attributed to the fluctuating quadrupole moment) is presented as the first theoretical derivation, yet the manuscript provides no visible step-by-step derivation, error analysis, or quantitative comparison to observed eccentricities. This leaves the support for the weak scaling and its explanatory power only moderately defensible.

minor comments (2)

Notation for masses (M for initial progenitor mass, m_wd for white dwarf mass, m_e for envelope mass) should be defined explicitly at first use in the abstract and consistently throughout to improve readability.
The abstract and discussion would benefit from a brief statement of the predicted eccentricity range or typical values from the model to allow direct comparison with observations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive review and positive assessment of the significance of our results. We address the major comments point by point below, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: The central claim that intermediate-mass systems detach at non-degenerate helium ignition (setting m_e and enabling the e ∝ m_e^{1/6} explanation) is load-bearing but rests on an assumption whose generality is not demonstrated. No comparisons are shown to full binary-evolution grids varying initial mass, metallicity, or mass-transfer efficiency to confirm the detachment epoch holds across the 3-5 M⊙ range.

Authors: We acknowledge that the timing of detachment at non-degenerate helium ignition for progenitors in the 3-5 M⊙ range is central to our analysis of m_e and the resulting eccentricity scaling. This timing follows from standard single-star evolution models for stars that develop non-degenerate helium cores. Our manuscript reproduces specific prior numerical binary calculations for representative systems in this channel and derives an analytical relation for the final orbital period under this assumption. To demonstrate broader applicability, we will add references to stellar evolution grids at varying metallicities and include a limited set of additional binary evolution calculations that vary mass-transfer efficiency for selected initial masses in the 3-5 M⊙ range. These additions will confirm that the detachment epoch remains consistent and will better support the generality of the m_e^{1/6} scaling. revision: yes
Referee: The eccentricity calculation (attributed to the fluctuating quadrupole moment) is presented as the first theoretical derivation, yet the manuscript provides no visible step-by-step derivation, error analysis, or quantitative comparison to observed eccentricities. This leaves the support for the weak scaling and its explanatory power only moderately defensible.

Authors: We agree that the eccentricity derivation would benefit from greater explicitness to fully substantiate the weak scaling and its implications. In the revised manuscript we will insert a clear step-by-step derivation of the eccentricity generated by the fluctuating gravitational quadrupole moment of the convective envelope at detachment, including the key equations and the origin of the m_e^{1/6} dependence. We will also add an assessment of uncertainties associated with the envelope fluctuation assumptions and a direct quantitative comparison between the predicted eccentricities and the observed values for millisecond pulsar systems with CO white dwarf companions. These changes will strengthen the support for the scaling and its ability to unify the eccentricity distributions across white dwarf types. revision: yes

Circularity Check

0 steps flagged

No significant circularity; eccentricity derivation is independent of fitted inputs

full rationale

The paper reproduces prior numerical binary-evolution calculations as external input to establish the detachment epoch at non-degenerate helium ignition for 3-5 M⊙ progenitors, then derives the eccentricity from the fluctuating gravitational quadrupole of the convective envelope at that epoch. The stated scaling e ∝ m_e^{1/6} follows directly from the quadrupole-fluctuation mechanism rather than being imposed by construction or by fitting to the target eccentricities. The supplementary analytical model for final orbital period P(M, m_wd) is presented as a simple relation extracted from the same reproduced calculations, without reducing the central eccentricity claim to a tautology or self-citation chain. No load-bearing self-citations, uniqueness theorems, or smuggled ansatzes appear in the provided text. The derivation therefore remains self-contained against external stellar-evolution benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard binary evolution assumptions about mass transfer stability and envelope dynamics, with the new analytical relation and eccentricity formula introduced without additional free parameters or invented entities visible in the abstract.

axioms (2)

domain assumption Stable Roche-lobe overflow occurs for intermediate-mass stars (3-5 solar masses) at the end of the main sequence or beginning of hydrogen shell burning.
This defines the formation channel under study.
domain assumption Eccentricity is determined by the fluctuating gravitational quadrupole moment of the convective envelope at detachment.
This is the physical basis for the new eccentricity calculation.

pith-pipeline@v0.9.0 · 5624 in / 1545 out tokens · 66787 ms · 2026-05-10T17:34:56.290894+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the eccentricity e in this process, which is set by the fluctuating gravitational quadrupole moment of the progenitor's convective envelope during Roche-lobe detachment... e∝m_e^{1/6}
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Intermediate-mass progenitors detach when their non-degenerate cores ignite helium

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

2013, Stellar Structure and Evolution, doi: 10.1007/978-3-642-30304-3

Chisabi M., et al., 2025, MNRAS, 537, 2462 Cohen Y ., Ginzburg S., Levy M., Bar Shalom T., Siman Tov Y ., 20 24, MNRAS, 534, 455 EPTA Collaboration et al., 2023, A&A, 678, A48 Eggleton P . P ., 1983,ApJ, 268, 368 Freire P . C. C., Tauris T. M., 2014, MNRAS, 438, L86 Freire P . C. C., et al., 2012, MNRAS, 423, 3328 Ginzburg S., Chiang E., 2022, MNRAS, 509,...

work page doi:10.1007/978-3-642-30304-3 2025