S301 and friends: Measuring the spin of Sgr A*

A. Burkert; B. Aytac; D. Calderon; D.C. Ribeiro; F. Eisenhauer; F. Mang; G. Bourdarot; H.B. Perets; J. Cuadra; M. Sadun Bordoni

REVIEW 3 major objections 5 minor 12 references

Stars with matching apocenters but larger pericenters can calibrate away Newtonian nodal precession so S301 isolates Sgr A*’s spin.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.5

2026-07-31 05:36 UTC pith:XEE57US4

load-bearing objection Clean celestial-mechanics strategy for S301: apocenter-matched stars can calibrate Newtonian nodal torque while LT stays pericenter-dominated; the open issue is residual accuracy under granularity, not the scaling itself. the 3 major comments →

arxiv 2607.24931 v1 pith:XEE57US4 submitted 2026-07-27 astro-ph.GA astro-ph.HE

S301 and friends: Measuring the spin of Sgr A*

T. Piran , P. Amaro-Seoane , B. Aytac , G. Bourdarot , A. Burkert , D. Calderon , J. Cuadra , F. Eisenhauer

show 10 more authors

R. Genzel S. Gillessen S. Joharle F. Mang T. Naab T. Ott H.B. Perets D.C. Ribeiro M. Sadun Bordoni R. Sari

This is my paper

classification astro-ph.GA astro-ph.HE

keywords Sgr A*Lense-Thirring precessionstellar orbitsGalactic Centerblack hole spinNewtonian confusionS301nodal precession

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.

S301 skims Sgr A* at only 280 gravitational radii, so Lense–Thirring nodal precession from the black hole’s spin is large enough to measure within a few orbits. The obstacle is Newtonian confusion: any flattened mass around the black hole also twists orbital planes. The paper shows that for a disk or flattened distribution that reaches beyond the stars’ closest approaches, the secular Newtonian torque on a highly eccentric orbit is set mainly by how far the star goes out (the apocenter), while the spin signal is set by how close it comes (the pericenter). S2, S55 and S38 share apocenters comparable to S301’s but have much larger pericenters, so they feel roughly the same Newtonian torques yet only ~1/30 of S301’s spin signal. Their measured precessions therefore map the mass and orientation of the extended matter; that background can be subtracted from S301, leaving the relativistic spin signature. Schwarzschild apsidal advance further rotates S301’s pericenter relative to any disk, imprinting a time-varying Newtonian piece while the spin piece stays fixed. With continued high-precision astrometry and spectroscopy, the in-plane spin projection may be reachable soon; the full spin vector needs a longer accumulation of the apsidal Lense–Thirring term.

Core claim

For a disk or flattened mass extending beyond the stellar pericenters, secular Newtonian nodal torque on a highly eccentric orbit is controlled mainly by apocenter, whereas Lense–Thirring scales as (rg/rp)^{3/2}. Stars with apocenters comparable to S301’s but much larger pericenters (S2, S55, S38) therefore experience comparable Newtonian torques while carrying ~30 times smaller LT signals, so their precessions calibrate the Newtonian background for subtraction from S301.

What carries the argument

Apocenter dominance of the secular Newtonian torque (analytic estimates validated by orbit-averaged numerical torques): in the embedded and crossing regimes the torque is set by material near the largest radii the orbit reaches, while LT is fixed by pericenter; this differential scaling, plus multi-star geometric triangulation of the unknown disk plane, enables the subtraction.

Load-bearing premise

The extended mass must be smooth enough and reach beyond the reference stars’ closest approaches; a contrived intermediate disk or a few massive granular perturbers can under-calibrate the subtraction or leave a stochastic floor that averaging only partly removes.

What would settle it

Joint multi-star fits of measured nodal precessions of S301 together with S2, S55 and S38: if the residual after Newtonian subtraction fails to match the predicted LT amplitude and orientation dependence, or if time variation with Schwarzschild-driven ω advance does not isolate a constant LT offset, the separation claim fails.

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

If this is right

In-plane spin projection of Sgr A* becomes measurable on a few-orbit timescale with continued GRAVITY+ and ELT data.
Full three-dimensional spin vector requires longer-term isolation of the LT apsidal term after Schwarzschild subtraction.
Existing non-detections on S2 already tighten the allowed Newtonian quadrupole interior to ~S2’s apocenter, reducing S301’s confusion ratio.
Mutually misaligned reference orbits triangulate disk mass and orientation in three dimensions; S29 anchors the radial profile.
Time dependence from Schwarzschild apsidal advance supplies an independent pure-disk diagnostic separable from the constant LT signal.

Where Pith is reading between the lines

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

If the method succeeds, Sgr A* becomes the first black hole with a directly measured spin vector from stellar dynamics rather than from accretion-flow modeling.
The same apocenter-matched multi-star subtraction can be ported to other galactic nuclei once stars with comparable apocenters and disparate pericenters are found.
Granularity simulations at S301’s pericenter will be required before claiming percent-level spin precision; residual Brownian recoil of the black hole itself may set a correlated noise floor in joint fits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Clean celestial-mechanics strategy for S301: apocenter-matched stars can calibrate Newtonian nodal torque while LT stays pericenter-dominated; the open issue is residual accuracy under granularity, not the scaling itself.

read the letter

The useful core is simple and solid. For disks or flattened mass that extend past the stellar pericenters, secular Newtonian nodal torque on a high-e orbit is set mostly by apocenter, while LT scales as (rg/rp)^{3/2}. S301 and the apocenter-matched set (S2, S55, S38) therefore feel comparable disk torques but very different LT signals (~30× for S2). That is the right lever for a joint subtraction program, and they back it with Gauss planetary equations, regime classification (embedded/crossing vs interior quadrupole), and orbit-averaged numerics that give disk-torque ratios of order unity across (i,ω).

What is actually new is not LT or disk precession in the abstract—those are in Will, Merritt, Iorio, Heißel—but the concrete S301-centered multi-star strategy, the apocenter-dominance argument applied to these elements, the time-variation discriminant from Schwarzschild ω-sweep, and the numerical disk-to-LT table for the real stars. Citations are appropriate; circularity is low (Menc bound comes from prior GRAVITY S2 work, not fitted here to force a spin claim). Fiducials (χ=1, Menc=10^3 M⊙, α=−1) are labeled as benchmarks.

Soft spots are real but mostly flagged by the authors. The intermediate-disk shell (280–2800 rg) undercalibrates if mass sits there; that configuration is contrived but not impossible. Granularity is the sharper limit: Sadun Bordoni-type ~100 M⊙ perturbers can bias smooth fits and set a stochastic floor that star/orbit averaging only partly kills, and they have not shown end-to-end recovery of χ with noise, A/B orbit ambiguity, common-mode Brownian recoil, and a joint fit. That is a calibration-accuracy gap, not a failure of the torque scaling. No code, no full mock—fine for a strategy paper, but referees will want a clearer residual budget.

This is for Galactic-center dynamicists and anyone planning GRAVITY+/ELT spin work. Worth a serious referee. I would engage: read it, cite the multi-star lever if I write on Sgr A* spin or cluster torques, and push for a granular end-to-end test next.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a multi-star strategy to isolate the Lense–Thirring (LT) nodal precession of the newly discovered star S301 (rp≈280 rg) from the Newtonian nodal precession induced by an extended mass distribution around Sgr A*. The key argument (§2, Eqs. 1–3) is that for any disk extending beyond the stellar pericenters, the secular Newtonian torque on a highly eccentric orbit is controlled by the apocenter, whereas the LT rate scales as (rg/rp)^{3/2}. Since S301, S2, S55 and S38 have comparable apocenters but pericenters differing by a factor ~10, the reference stars experience Newtonian torques of the same order while their LT signals are ~30× smaller, so their measured precessions can calibrate the Newtonian background for subtraction from S301. The analytic estimates are validated by orbit-averaged numerical torque calculations for an α=−1 disk (Table 1, Fig. 4), and the paper adds two further discriminants: Schwarzschild-driven time variation of the disk term (§8) and the apsidal combination carrying χcosβ (§6). Granularity of the perturber population is acknowledged (§7) as a stochastic floor.

Significance. If the strategy holds, it offers a credible near-term path to the first direct measurement of the in-plane spin projection of Sgr A* — a measurement no other black hole permits — using only existing or imminently available stars and instruments (GRAVITY+, ELT). The core scaling argument is essentially parameter-light: it follows from the Gauss planetary equation plus standard LT scaling, with fiducial choices (χ=1, Menc=10^3 M⊙, α=−1) stated as benchmarks rather than fitted, and the razor-thin disk is shown to be a conservative upper bound on the confusion (§4, spheroid test). The mechanism is validated by independent numerical torque integrations, and the paper ships falsifiable discriminants — the ω-dependent time variation of the disk term versus the constant LT offset (§8), and the nodal/apsidal ratio that cancels χ (Eq. 16). The manuscript is also commendably candid about its weakest point (granularity, §7).

major comments (3)

[§2, Eq. (2); §3, Eqs. (6)–(7); §4, Table 1] §2, Eq. (2) and §3, Eqs. (6)–(7): the function g(e,α) carries the apocenter-dominance claim, but it is never given explicitly. For α=0 it diverges as (1−e)^{-1/2} (a factor ~7.6 for S301), 'cut off in practice' by the disk inner edge or finite thickness; for α=−1 the eccentricity dependence 'nearly cancels'. All quantitative confusion ratios (Eq. 7, Tables 1–2, Fig. 4) are shown only for α=−1 — the profile the paper itself identifies as minimizing the disk confusion. §4 states 'similar results are obtained for other values of α', but these are not shown. Please provide the explicit form (or derivation) of g(e,α) with the cutoff prescription, and a numerical table/figure for α=0 and a steeper profile, plus a physical justification that α=−1 (Mestel) is representative of the actual stellar cusp rather than a best case.
[§5, Eqs. (13)–(14)] §5, Eqs. (13)–(14): the central deliverable is a joint fit, but the manuscript does not show that the subtraction residual on S301 can be driven below ΔΩ_LT^S301. Three unquantified degradations compound: (i) the disk plane is a free parameter, so the reference stars' true (i,ω) differ from S301's and the torque ratio fluctuates over 0.26–1.0 (Table 1); (ii) at (i=10°, ω=90°) S301 itself is disk-dominated (ratio 2.5), so recovery leans entirely on the differential signal; (iii) Sgr A*'s Brownian recoil (§7) enters a joint astrometric fit as correlated, common-mode noise that star-averaging does not suppress. A Fisher-level or Monte-Carlo error budget — even in the smooth-disk case with realistic GRAVITY+/ELT precisions — is needed to support the conclusion that the in-plane spin 'may be within near-term reach'; otherwise that claim should be tempered.
[§7] §7: the granularity evidence (Sadun Bordoni et al. 2025) is for S2; at S301's 280 rg pericenter the relevant perturber count, the fixed-cusp approximation, and the stochastic floor are all unestablished, as the paper concedes. Two further points deserve treatment: (a) the same perturber realization acts on all calibration stars, so ensemble-averaging over reference stars suppresses the floor less than independent-realization averaging would imply; (b) a bias of a smooth-mass fit 'by up to a factor ~6' (at S2) would translate directly into a mis-subtraction on S301 if unmodeled. Since the near-term detection claim rests on this floor being below ~10^-4 rad/orbit, either a quantitative estimate at S301's pericenter or explicit conditioning of the conclusions on the smooth-background assumption is needed.

minor comments (5)

[§9] §9, Conclusions: the inference that S2's nodal non-detection implies a quadrupole 'a factor of ~3 smaller' than the maximal disk assumes order-unity geometric factors and the α=−1 normalization; please state these assumptions explicitly, as the bound is geometry-dependent per Table 1.
[§3] §3: the angles β and λ and the frame dependence of the nodal rates are defined densely; a short summary table of the three LT observables and the spin components they measure (nodal: χsinβsinλ; inclination: χsinβcosλ; apsidal: χcosβ) would aid the reader, as this structure is reused in §§5–6.
[Table 2] Table 2: the 'Strategy' column mixes literature citations [1]–[5] with methodological remarks; consider separating references from strategy text. Also 'Zero first order; acts as orbital clock' for the Schwarzschild row is cryptic without reference to §8.
[throughout] Typesetting: the collaboration name appears as 'GRA VITY' throughout (a macro spacing issue); '1◦.95' should be 1.95°; Table 3's header alignment is broken ('Semi-Major Axis (a)' spans the eccentricity column); S29 is introduced abruptly in §1 ('the wider orbit of S29 anchors the radial profile') before its parameters appear in Table 3.
[Fig. 4] Fig. 4 caption: 'the ratios vanish at i=90°, where the disk torque ∝ cos i goes to zero' — note that the plotted range stops at 75°, so this statement describes an extrapolation; either extend the curves or move the remark to the text.

Circularity Check

0 steps flagged

No significant circularity: the apocenter/pericenter torque separation follows from standard Gauss/LT formulae plus explicit orbit-averaged numerics, not from fitted or self-defined inputs.

full rationale

The load-bearing chain is: (i) Gauss planetary equation for nodal rate from an axisymmetric disk potential (Brouwer & Clemence; Murray & Dermott); (ii) standard LT nodal/inclination rates (Will 2008; Lense & Thirring); (iii) an analytic argument that exterior-ring torque and Kepler time-weighting make secular Newtonian nodal precession apocenter-dominated when r_disk ≳ r_p, versus LT ∝ (r_g/r_p)^{3/2}; (iv) numerical orbit-averaged torque integrals that validate order-unity disk-torque ratios for S301 vs S2/S55/S38 (Table 1, Fig. 4). Fiducial M_enc = 10^3 M_⊙ and χ = 1 are stated benchmarks taken from prior GRAVITY apsidal bounds and maximal-spin convention, not parameters fitted inside this paper to force a spin detection. S301 orbital elements and the Menc upper limit are observational inputs from GRAVITY Collaboration papers, not theoretical uniqueness claims. Overlapping-author citations (S301 discovery; Sadun Bordoni et al. 2025 on granularity) supply data or caveats and do not close a definitional loop on the torque-scaling result. The paper proposes a calibration strategy rather than claiming a measured χ; nothing in the derivation reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The load-bearing content rests on standard celestial-mechanics and post-Newtonian formulas plus observational inputs from GRAVITY. Fiducial numbers (Menc, χ, α, rdisk) are chosen by hand as benchmarks inside published bounds; they scale the confusion ratios but do not define the apocenter-vs-pericenter separation. No new physical entities are postulated. The main domain assumptions are the existence of an extended flattened mass that reaches beyond the reference-star pericenters and that granularity remains a suppressible floor rather than a show-stopper.

free parameters (5)

Menc (fiducial disk mass) = 10^3 M⊙
Set to 10^3 M⊙ as fiducial inside the GRAVITY Collaboration (2024) upper bound Menc ≲ 1200 M⊙; scales all Newtonian rates linearly and sets the confusion ratios in Eqs. 6–8 and Table 1.
χ (dimensionless spin) = 1
Adopted as maximal spin χ=1 throughout; LT rates scale linearly with χ, so smaller spins are proportionally harder. Not fitted to data in this work.
disk surface-density index α = −1 (fiducial)
Primary numerical results use α=−1 (Mestel); analytic discussion also covers α=0. Choice affects g(e,α) and the (1−e) factor in confusion ratios but not the apocenter-dominance regime classification.
disk outer radius rdisk = r_a(S301) in Table 1
Fiducial numerics place rdisk at S301’s apocenter; regime analysis scans embedded/crossing/interior cases. Controls which torque regime applies.
disk scale height H = 0.01 r′
Softening H=0.01 r′ used to regularize disk-crossing divergences in the numerical potential average; a numerical regularization choice.

axioms (6)

standard math Gauss planetary equation for nodal rate dΩ/dt = −(1/(h sin i)) ∂⟨Φdisk⟩/∂i
Invoked as Eq. 1 (§2); classical result from Brouwer & Clemence / Murray & Dermott.
standard math Lense–Thirring nodal and inclination rates (Eqs. 4, 9) and node-corrected apsidal combination Δϖ_LT ∝ χ cos β
Standard PN frame-dragging formulas (Lense & Thirring 1918; Barker & O’Connell 1975; Will 2008), used throughout §§3 and 6.
domain assumption Orbit averaging of the disk potential with Kepler time weight dt ∝ r² df in the fixed SMBH background (Mdisk ≪ MBH)
Stated in §2 and Eqs. 10–11; required for the secular torque approximation.
domain assumption Among axisymmetric mass distributions with fixed radial profile, a razor-thin disk maximizes the quadrupole and hence Newtonian nodal precession
Used to treat disk results as upper limits on confusion (§1 and end of §4); supported by the paper’s own oblate-spheroid scan but still an idealization of the real cluster.
domain assumption Extended mass around Sgr A* satisfies Menc ≲ 10^3 M⊙ inside S2’s orbit, from prior S2 apsidal precession
Taken from GRAVITY Collaboration (2020, 2024) and used as the mass scale for all confusion estimates (§1).
ad hoc to paper For disks extending beyond stellar pericenters, secular Newtonian torque is apocenter-dominated (embedded/crossing regimes)
Core modeling claim derived in §2; load-bearing for the multi-star calibration. Holds under the stated radial-extension assumption and is the premise the strategy rests on.

pith-pipeline@v1.2.0-grok45-kimik3 · 20761 in / 4401 out tokens · 88113 ms · 2026-07-31T05:36:47.113796+00:00 · methodology

0 comments

read the original abstract

The discovery of S301 (GRAVITY Collaboration et al., 2026) with pericenter distance rp= 280rg and eccentricity e=0.9825, opens the prospect of measuring the spin parameter of Sgr A* through Lense--Thirring (LT) nodal precession. A major obstacle is Newtonian confusion: any non-spherical extended mass distribution can also induce nodal precession. We aim to separate the LT spin signal of S301 from the Newtonian nodal precession. We compare the secular Newtonian torque exerted by a disk or flattened mass distribution on the orbits of S301 and of the apocenter-matched reference stars S2, S55, and S38, using analytic estimates validated by numerical orbit-averaged torque calculations. For a disk or flattened distribution extending beyond the stellar pericenters, the secular Newtonian torque on a highly eccentric orbit is controlled mainly by the apocenter, whereas the LT signal is controlled mainly by the pericenter. Thus stars with apocenters comparable to S301's but much larger pericenters, in particular S2, but also S55, and S38, experience comparable Newtonian torques while having ~ 30 times smaller LT signals (for S2). Their measured precessions, or upper limits on them, can therefore calibrate the mass and orientation of the Newtonian background for subtraction from S301's precession. The Schwarzschild apsidal advance of S301 further rotates the orbit's pericenter relative to any disk, producing a systematic time dependence in the Newtonian contribution, while the LT signal remains fixed by the spin vector. Granularity of the perturber population sets a stochastic floor on this subtraction, which orbit- and star-averaging suppress. With continued GRAVITY+ astrometry and Extremely Large Telescope (ELT) spectroscopy, the in-plane spin projection may be within near-term reach; the full spin vector requires a much longer-term accumulation of the LT apsidal signal.

Figures

Figures reproduced from arXiv: 2607.24931 by A. Burkert, B. Aytac, D. Calderon, D.C. Ribeiro, F. Eisenhauer, F. Mang, G. Bourdarot, H.B. Perets, J. Cuadra, M. Sadun Bordoni, P. Amaro-Seoane, R. Genzel, R. Sari, S. Gillessen, S. Joharle, T. Naab, T. Ott, T. Piran.

**Figure 1.** Figure 1: Orbits of the benchmark S-stars and S301 around Sgr A* in gravitational radii, projected in 3D. Left: full-scale view; the apocenters set the scale. Right: zoom into the inner ∼ 6000 rg, showing the disparate pericenters and the strongly inclined, mutually misaligned orbital planes. S301 is shown for its two degenerate orbit solutions (A, B). The inner tracers S301, S55, S2, and S38 have comparable apocent… view at source ↗

**Figure 2.** Figure 2: The three orbit–disk configurations, shown edge-on for the same inclined eccentric orbit (drawn with e = 0.75 and i = 35◦ for clarity; the black circle marks the black hole, red and green dots the pericenter and apocenter): (a) embedded, ra ≲ rdisk; (b) crossing, rp ≲ rdisk ≲ ra ; (c) interior quadrupole, rdisk ≲ rp. In (a) and (b) the secular nodal torque is dominated by the apocenter, while in (c) it is … view at source ↗

**Figure 3.** Figure 3: Decomposition of the orbital-pole drift. Any torque rigidly tilts the orbital plane, producing a frame-independent drift dLˆ of the orbital pole Lˆ , which sits at colatitude i from the reference-plane normal ˆz. The drift decomposes into a meridional part δi , which changes the inclination directly (di = δi ; Eq. 9), and an azimuthal part δΩ, which moves the node along a circle of radius sin i, so that ∆… view at source ↗

**Figure 4.** Figure 4: Disk-to-LT confusion ratio |∆Ωdisk/∆ΩLT| as a function of the orbit–disk inclination i, for S301 (blue) and S2 (red) and for ω = 0 ◦ (solid, circles) and ω = 90◦ (dashed, squares), computed for the α = −1 disk of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

discussion (0)

Reference graph

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