Rediscovering the Milky Way with orbit superposition approach and APOGEE data I. Method validation

Bridget Ratcliffe; Davor Krajnovic; Glenn van de Ven; Ivan Minchev; L\'ea Marques; Marica Valentini; Matthias Steinmetz; Misha Haywood; Nikolay Kacharov; Paola Di Matteo

REVIEW 3 major objections 6 minor 2 cited by

Rediscovering the Milky Way with orbit superposition approach and APOGEE data I. Method validation

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A stellar-orbit superposition method can reconstruct the Milky Way's density, kinematics, and metallicity across the whole galaxy from APOGEE-like samples alone.

desk verdict Solid method validation with an honest design, but the claim that APOGEE data alone suffice is not yet supported—no test perturbs the assumed potential. read the letter →

arxiv 2411.15062 v1 pith:BZMV76V5 submitted 2024-11-22 astro-ph.GA

Sergey Khoperskov , Glenn van de Ven , Matthias Steinmetz , Bridget Ratcliffe , Ivan Minchev , Davor Krajnovic , Misha Haywood , Paola Di Matteo

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Nikolay Kacharov Léa Marques Marica Valentini Roelof S. de Jong

This is my paper · ORCID

classification astro-ph.GA

keywords orbitsuperpositionSchwarzschildmethodMilkyWaystructureAPOGEEsurveyselectionfunctionbarredgalaxystellarmetallicitydistributiongalactickinematics

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 method-validation paper asks whether a Schwarzschild-style orbit superposition model can turn the patchy, Sun-centred sampling of a Milky Way-like galaxy into a complete picture of the whole disc. Using mock catalogues that mimic the APOGEE DR17 footprint drawn from a simulated barred galaxy, the authors show that orbits of observed stars, weighted to reproduce an adopted 3D stellar density, recover the galaxy's density, velocity distributions, energy–angular momentum structure, circularity, and global metallicity distribution even outside the surveyed area. The test succeeds for a giant-star sample (Mock 1), reconstructing velocity dispersions to about 10 km/s and all three peaks of the metallicity distribution, while a red-clump-like sample fails in the inner galaxy and produces an artificial spheroid. The paper's point is that present-day APOGEE data are sufficient to derive selection-function-corrected, whole-galaxy chemo-kinematic maps, provided the gravitational potential is trusted.

What carries the argument

The machinery is the Schwarzschild orbit superposition method adapted to the Milky Way and executed in the bar's rotating reference frame. A gravitational potential is built from the simulation snapshot via spherical-harmonic expansion with $l_{\max}=m_{\max}=20$, each star in the mock sample is integrated as an orbit for 5 Gyr in that potential, and a non-negative weight $w_i$ is assigned to each orbit so that $\rho_s(r)=\sum_i w_i\rho_i(r)$ matches the adopted 3D stellar density on a $50^3$ Cartesian grid. Weights come from least-squares fits to random 10% orbit subsets, averaged over ten realisations to combat degeneracy. Because the frame rotates with the bar, a star's orbit represents all unobserved stars sharing that orbit at the present epoch, which is what lets the model paint [Fe/H] and kinematics beyond the survey footprint.

What would settle it

Build the same mock catalogues but integrate the orbit library in a potential deliberately different from the snapshot, for instance a bar pattern speed shifted by 20% or a bar angle rotated by 10 degrees, and compare the reconstructed density, kinematics, and metallicity distribution to the snapshot truth. The paper's Section 4.1 predicts visible failure, and a clean measurement of the error threshold for each input perturbation would confirm or refute the method's practical promise.

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Extended reading notes

Core claim

The paper presents an orbit-superposition method and argues that it can stand in for the full distribution function of the Milky Way: each observed star is treated as a tracer of a whole family of unobserved stars on the same orbit, so the orbit library can be weighted to reproduce the adopted stellar density, and stellar labels such as [Fe/H] can be moved along orbits to places no survey has seen. Validated on a simulated barred Milky Way, the giant-star-based Model 1 reproduces the adopted 3D density to within about 5%, recovers the bar's butterfly velocity pattern and the bar-resonance ridges in energy–angular momentum space, and matches the global metallicity distribution function including the metal-rich peak that is under-sampled in the mock catalogue. The red-clump-based Model 2 fails exactly where its input lacks stars with apocentres in the inner galaxy, producing an artificial, slowly rotating spheroid; the paper reads this failure as diagnostic of an incomplete orbital library rather than as a defect of the method itself.

Load-bearing premise

The load-bearing premise is that the Milky Way's gravitational potential, including the bar's pattern speed and orientation, is known well enough to integrate the orbits; the real potential is not known, and the paper concedes that a wrong potential would produce odd kinematics everywhere.

Editorial extensions

If this is right

Real APOGEE DR17 giant-star data can be projected into a full 3D chemo-kinematic map of the Milky Way, correcting the selection function without building an explicit completeness model.
The method transfers any stellar label tied to orbits, including metallicity and in principle ages and elemental abundances, to regions outside the survey footprint.
Residuals between the equilibrium orbit-superposition model and real data isolate non-equilibrium structures such as spiral arms, the warp, and phase-mixing signatures, since the model cannot generate them.
A sample that lacks stars whose apocentres lie in the inner galaxy will produce an artificial central spheroid, so future surveys should ensure inner-galaxy coverage.
Because the recovered kinematics depend on the adopted potential, matching bar-resonance features offers a route to constrain the bar's pattern speed and orientation.

Reading between the lines

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

Beyond the paper, the same weighted-orbit machinery could combine Gaia 6D data with asteroseismic ages to produce a self-consistent age–metallicity–kinematic atlas of the Milky Way.
Beyond the paper, the paper's diagnostic that a very broad weight distribution signals a degenerate solution gives a cheap sanity check for any future orbit-superposition map of the Galaxy.
Beyond the paper, a direct test of the potential assumption would rerun the mock with the bar pattern speed deliberately offset; the error threshold at which the reconstructed kinematics break would quantify how well the real potential must be known.
Beyond the paper, the finding that APOGEE's footprint alone creates a high-energy, low-rotation blob in E–Lz space warns that merger-debris searches in this space must model survey selection before claiming accretion signatures.

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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

The paper is a well-structured method validation with an honest design, but the headline claim that APOGEE DR17 is already sufficient for this kind of reconstruction goes beyond what the tests establish. The potential, bar pattern speed, and bar orientation all come from the same simulation snapshot and are never varied, so the central validation is self-referential in a way the authors themselves concede in Section 4.1.

What is new: applying Schwarzschild-style orbit superposition in the rotating bar frame to the Milky Way, using the orbits of observed stars as the basis, and then transferring chemical labels along orbits to correct for selection functions. Wylie et al. did unweighted orbit superposition; here the weights are fit to density only. The validation is set up as a genuine prediction: weights from density alone, then kinematics and [Fe/H] compared to the snapshot. Model 1 recovers the density within about 5%, velocity dispersions within about 10 km/s, and the main MDF peaks. The authors also deliberately show Model 2 failing and diagnose the artificial spheroid as arising from incomplete orbital families. That is honest and useful.

Soft spots: the benchmark is a single simulated galaxy, the mock data are noiseless, and the APOGEE-like selection is a HEALpix footprint approximation rather than the full magnitude and color target selection. Most importantly, no sensitivity test varies the potential. The paper says in Section 4.1 that a non-reliable potential will likely produce odd kinematics, but it never quantifies how quickly the reconstruction degrades as the bar pattern speed or halo parameters shift. That makes the summary bullet on APOGEE sufficiency a feasibility statement, not a validated result. Also, no code or data are released, which is a drawback for a methods paper.

One smaller point: the MDF recovery looks surprisingly good even in the failing Model 2. That suggests the global MDF test is less discriminating than the density and kinematics tests, so the metallicity claims should be read with that in mind.

The citation pattern is fine, and the limitations section is more candid than most. This paper will be useful for anyone working on selection-function corrections or chemo-kinematic mapping with upcoming surveys. It deserves a serious referee, because the method is promising and the validation design is mostly sound, but the revision should add at least a perturbed-potential test or a second simulation, and the claims about present-day data should be toned down accordingly.

Referee Report

3 major / 6 minor

Summary. The paper develops a Schwarzschild-style orbit superposition method for the Milky Way: stars with 6D phase-space coordinates from an APOGEE-like mock sample are integrated as orbits in an assumed Galactic potential, and non-negative orbit weights are fitted so that the superposition reproduces a prescribed 3D stellar density distribution. Kinematics, energy-angular momentum structure, circularity, and [Fe/H] distributions are then predicted without using those observables in the fit. The method is validated on a snapshot of an isolated N-body/hydrodynamical simulation of a barred MW-like galaxy, using two mock samples designed to mimic the APOGEE giant (Mock 1) and red-clump (Mock 2) footprints. Model 1 recovers the adopted density and reproduces the simulated kinematics and metallicity structure across most of the galaxy, while Model 2 fails in the inner regions and produces an artificial spheroidal component. The authors conclude that the orbit superposition approach can correct survey selection functions and that present-day APOGEE-like data are already sufficient for such modeling.

Significance. If the validation withstands scrutiny, the method offers a novel way to deproject the complex APOGEE footprint and to map chemo-kinematic structure over the whole Galaxy, including regions outside the survey area. The validation design is honest in an important respect: orbit weights are fitted only to the adopted stellar density, while kinematics and [Fe/H] are genuine predictions. The paper also reports the failure of Model 2 in detail, interprets it through incomplete orbital families, and uses ten weight realizations to mitigate the degeneracy of the non-orthogonal orbit library. These are real strengths. The principal limitation is that the benchmark assumes a known potential and a target density derived from the same simulation, so the extrapolation to real APOGEE DR17 data is not yet quantitatively supported; the paper itself flags the unknown MW potential in Section 4.1 but does not test its sensitivity.

major comments (3)

[Sec. 4.1 / Eq. (2)] The validation is performed entirely in a known-potential regime. The orbit library is integrated in the spherical-harmonic potential reconstructed from the same N-body snapshot (Sec. 2.1), the target stellar density is derived from that same potential through Eq. (2), and the bar pattern speed (22 km/s/kpc) and orientation (27 deg) are taken directly from the simulation. Section 4.1 states that the true MW potential is not known and that a non-reliable potential would produce odd kinematic features, but the paper presents no experiment that perturbs the potential, bar pattern speed, bar orientation, or halo/disc decomposition. Because a wrong potential changes both the orbital families and the density that the weights are forced to reproduce, the Summary bullet 5 claim that APOGEE DR17-like data are already sufficient for a high-quality orbit-superposition analysis of the MW is not yet quantitatively established. I request a sensitivity analysis, for example varying the bar pattern speed over the published MW range, changing the bar orientation by its uncertainty, and rescaling the dark halo contribution, with the degradation in recovered density, kinematics, and MDF quantified.
[Sec. 2.2 / Sec. 3.3] The mock catalogues use exact 6D phase-space coordinates from the simulation particles, with no distance, proper-motion, or radial-velocity uncertainties applied. Real APOGEE data rely on astrometric or photometric distances and spectroscopic velocities whose errors propagate into the initial conditions for orbit integration and into the density binning on the 3D grid. The paper's final claim is that present-day APOGEE data are sufficient (Summary bullet 5), but the validation does not include a mock with errors typical of APOGEE+Gaia, such as a few percent distance errors and a few km/s velocity errors. The test also uses a single snapshot of one isolated galaxy, so the sensitivity to the galaxy's evolutionary state is not explored. Adding at least one error-injected mock and, ideally, a second simulation snapshot would substantially strengthen the practical conclusion.
[Sec. 3.1 / Fig. 4] The ten weight realizations are shown to produce nearly identical reconstructed surface density, which motivates the use of averaged weights, but the paper does not quantify the scatter of the predicted kinematics (velocity moments, E-Lz density, circularity, and [Fe/H] maps) across these realizations. Because the orbit library is non-orthogonal and the weight solution is acknowledged to be degenerate, different weight sets can in principle yield the same density but different velocity structure. The agreement shown in Figs. 7-13 is for the averaged weights only; a plot of, e.g., the r-Vphi maps or velocity-dispersion profiles from the individual realizations with their spread would directly demonstrate that the kinematic predictions are robust. This is a necessary internal-consistency check for a method whose main added value is the prediction of kinematics and abundances.

minor comments (6)

[Fig. 6 caption] The caption says "Mock 1 (blue) and Mock 2 (blue; see details in Sec. 2.2)"; the second color should presumably be red, matching the text and other figures.
[Sec. 3.1] The notation "r = {x_j, y_j, z_j}_{j=1,...,50^3}" appears as "j=1,...,503"; please correct the exponent and clarify how the 500 time samples per orbit are mapped onto the 50^3 Cartesian grid.
[Sec. 5] In the sentence "The orbital weights represent the stellar mass-weighed complete distribution function," "weighed" should be "weighted."
[Sec. 2.2] The description of the mock selection would benefit from specifying which APOGEE DR17 release and which target-selection criteria (magnitude limits, extinction, S/N) are approximated by the HEALpix distance distribution; as written, the sentence "qualitatively reproduce" leaves the selection function under-specified for exact reproducibility.
[Eq. (3)] The summation "NX_i" is typeset ambiguously; it should be \sum_{i=1}^N w_i \rho(r)_i.
[Sec. 4.2] The survey name "WEA VE" should be "WEAVE."

Circularity Check

1 steps flagged · score 2.0 of 10

Density recovery is enforced by the fit, but the kinematics and metallicity maps are genuine predictions of that fit, so circularity is minor.

fitted input called prediction [Section 2, Eq. (3); Section 3.2, Fig. 5 (rightmost panels)]
"ρs(r) = Σ_i w_i ρ(r)_i (3) ... The solution of Eq. 3 provides us with the weights wi for the orbit of each star in the observational sample whose superposition results in the adopted stellar density. ... Model 1 exhibits a striking resemblance to the approximated stellar density, a similarity further emphasized in the residual density map depicted in the rightmost panel where the relative difference between the two densities is less than 5%."

The stellar density ρs(r) is not an output discovered from the data: it is an input adopted from the same simulation snapshot (Section 2.1), and the orbit weights are solved precisely to make Σ_i w_i ρ(r)_i equal it. Therefore the reported 'recovery' of the stellar density, including the <5% residual, is a measure of the least-squares fit to the target density, not an independent reconstruction. This step is circular by construction. The paper's more meaningful validation is kinematic and metallicity comparison against snapshot data, since those quantities are not used in the weighting, which is why the overall circularity is limited.

full rationale

The paper's central validation is a method test: can an APOGEE-like, spatially incomplete sample of 6D phase-space measurements, weighted to reproduce an adopted stellar density, recover the full phase-space distribution and the chemo-kinematic properties of a simulated galaxy? The density-matching step is by construction: Eq. (3) defines the solution as weights whose superposition equals the adopted ρs(r), and the paper itself calls it the 'adopted' density. Thus the density comparison in Section 3.2 is a fit diagnostic rather than a prediction. However, the kinematics (Section 3.3), E-Lz structure (Section 3.4), circularity (Section 3.4), and metallicity distributions (Section 3.5) are not used to determine the weights; they are computed by projecting the fitted orbital weights onto the simulated snapshot data, and the comparisons against the ground-truth snapshot are genuine and non-trivial. The benchmark is self-referential in the weaker sense that the potential, target density, and orbital library all derive from the same N-body snapshot, and the paper explicitly acknowledges in Section 4.1 that the real Milky Way potential is not known and that a non-reliable potential would produce odd kinematic features. That is a limitation of the validation, not a circular derivation. No load-bearing self-citation chain or imported uniqueness theorem is present. Overall, the kinematic and metallicity predictions have independent content, so the circularity score is low.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on physical assumptions about the Milky Way (known potential, equilibrium, representative orbital library, abundance-orbit coupling) plus implementation choices such as expansion order, grid resolution, pattern speed, and weight-averaging. No new particles, forces, or entities are introduced.

free parameters (7)

Orbit weights w_i = Non-negative least-squares solution, averaged over 10 realizations
Variables of Eq. 3, fitted to the target stellar density. There are roughly 40,000 orbit weights, so the density recovery is enforced rather than predicted.
Bar pattern speed Omega_b = 22 km/s/kpc
Instantaneous value measured from the simulation snapshots, used to define the rotating frame in Section 3.1. No sensitivity test is performed, and the MW value is uncertain.
Spherical-harmonic expansion order = lmax = mmax = 20
Chosen in Section 2.1 to resolve the bar and X-shape while excluding spiral arms, which affects both the potential and the target density.
Density grid resolution = 50^3 bins over a 30 kpc cube
Resolution choice used to project orbits and the target density in Section 3.1; affects reconstruction accuracy and computational cost.
Orbit subset fraction = 10%
Selected after experiments over the 2-30% range as the best compromise between stability and cost; affects the weight distribution and degeneracy.
Orbit integration time = 5 Gyr, 500 points per orbit
Chosen in Section 3.1 to sample each orbit; not varied or justified from convergence tests.
Number of weight realizations = 10
Weights are averaged over ten random-subset solves to mitigate degeneracy; the number is chosen by the authors.

assumptions (6)

domain assumption The total gravitational potential and the stellar density of the galaxy are known or well constrained independently of the reconstructed kinematics.
Section 2, Eqs. 1-2 use this to define the target density; Section 4.1 admits the true MW potential is not known and that results are governed by the adopted potential.
domain assumption The galaxy is in dynamical equilibrium with a steadily rotating bar at a single constant pattern speed.
Section 3.1 integrates orbits in a rotating frame with an instantaneous bar pattern speed; Section 4.1 acknowledges that spiral arms, warp, bar slowdown, and buckling are not captured.
domain assumption The input stellar sample spans all orbital families needed to represent the target density.
Model 2 fails because its orbital library is incomplete, as discussed in Sections 3.2, 3.3, and the Summary; the method's success depends on this coverage.
domain assumption Stellar chemical abundances are evolutionarily coupled to orbital parameters, so transferring [Fe/H] labels along orbits is valid.
Section 3.5 relies on this coupling to reconstruct metallicity maps from the weighted orbit library.
domain assumption Mirroring each orbit along the principal axes preserves the distribution function and does not introduce artifacts.
Section 3.1 applies mirroring to enforce symmetry; the authors say other symmetries made no substantial difference, but no quantitative comparison is shown.
domain assumption Mock APOGEE samples built from the APOGEE distance distribution per HEALpix cell adequately represent the real APOGEE DR17 selection function.
Section 2.2 constructs Mock 1 and Mock 2 this way, without modeling magnitude limits, extinction, or target-allocation incompleteness in detail.

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Cite this review

Pith. "Pith review of Rediscovering the Milky Way with orbit superposition approach and APOGEE data I. Method validation." pith.science (2026). https://pith.science/paper/BZMV76V5

@misc{pith2026241115062,
  author       = {Pith},
  title        = {Pith review of: Rediscovering the Milky Way with orbit superposition approach and APOGEE data I. Method validation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BZMV76V5}},
  note         = {Machine review of arXiv:2411.15062}
}

read the original abstract

We introduce a novel orbit superposition method designed to reconstruct the stellar density structure, kinematics, and chemical abundance distribution of the entire Milky Way by leveraging 6D phase-space information from its resolved stellar populations, limited by the spatial coverage of APOGEE DR17.

Figures

Figures reproduced from arXiv: 2411.15062 by the authors.

**Figure 1.** Example of two orbits (red and blue) of disc stars in a barred potential in inertial (left) and rotating with the bar (right) reference frames. In the bottom panels, the orbits are colour-coded to indicate the progression of time. The ovals in the figure represent the bar orientation, with three different configurations colour-coded to denote changes in its orientation over time in the inertial frame. Notably, th… view at source ↗

**Figure 2.** Approximation of the stellar disc density from the simulation. The surface density of star particles in a snapshot is shown in the left panel. In the middle panel, we present the surface density of the stellar disk obtained from the potential approximated using multipole expansion (see Section 2.1 for details). The rightmost panel shows the difference between the two maps, divided by the snapshot stellar density. No… view at source ↗

**Figure 3.** Initial selection of star particles from the simulation mimicking the APOGEE footprint. In the left panel, the surface density of all star particles in the snapshot is shown. The middle and right panels show two mock catalogues which qualitatively reproduce the APOGEE DR17 giants (Mock 1) and red clump (Mock 2) stars footprints, respectively (see Sec. 2.2 for details). The number of star particles in the snapshot is… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Analysis of weights of orbits across various orbit superposition model realizations using Mock 1. The five left panels of the top row illustrate the weight distribution in five realizations of the orbit superposition using different 10% randomly sampled orbits, while t…

**Figure 5.** Figure 5: Results of the orbit superposition method. The left panels display the stellar density of orbits with constant weights, essentially presenting the stacked library of orbits of all star particles from Mock 1 (top) and Mock 2 (bottom). The middle panels depict the stella…

**Figure 6.** Figure 6: Mean weights distribution in two orbit superposition models obtained in ten realizations for the APOGEE-like Mock 1 (blue) and Mock 2 (blue; see details in Sec. 2.2). The weight values are given in relative units. The broad distribution of weights in Model 2 indicates…

**Figure 7.** Figure 7: Distribution of velocity components (VR in left, Vϕ in the middle and Vz in the right) in the galactic rest frame within three 2 kpc-wide annuli centred at 1 (top), 8 (middle), and 11 kpc (bottom). The snapshot data are represented by the grey-filled areas, while the r…

**Figure 8.** Figure 8: Reconstruction of the rotational velocity distribution as a function of the galactocentric distance. The left panel displays the stellar density distribution (normalized by the total stellar mass) in the simulation, while the middle and right panels depict the stellar …

**Figure 9.** Figure 9: Comparison of the face-on kinematics in the simulation snapshot and orbit superposition Model 1. From top to bottom: the mean velocities VR, Vϕ, Vz , and velocity dispersion components σR, σϕ, σz . The left column corresponds to the snapshot data, the middle one depic…

**Figure 10.** Figure 10: Reconstruction of energy-angular momentum space. The top row displays the initial distribution of stars from the simulation in the E − Lz space in two APOGEE-like mock spatial selections adopted for analysis. The bottom row presents a comparison between the snapshot d…

**Figure 11.** Figure 11: Reconstruction of circularity distribution as a function of the galactocentric distance. Similar to [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: The left panel displays the [Fe/H] distribution for stars in the initial model selections (blue and red). The right panel shows the results of the orbital decomposition. In both panels, the MDF of stars in the simulation snapshot is represented by the grey-filled his…

**Figure 13.** Figure 13: Reconstruction of the face-on stellar disc metallicity maps. The top panels display the mean [Fe/H] maps for the snapshot data (left) and the orbit superposition modelling based on Model 1 (middle) and Model 2 (right). The bottom panels illustrate the residuals betwee…

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Rediscovering the Milky Way with orbit superposition approach and APOGEE data II. Chrono-chemo-kinematics of the disc
astro-ph.GA 2024-11 conditional novelty 6.0 of 10

Orbit superposition of APOGEE stars yields a mass-weighted Milky Way disc map showing a less prominent alpha-bimodality and two distinct inner and outer age-metallicity sequences.
Rediscovering the Milky Way with orbit superposition approach and APOGEE data III. Panoramic view of the bulge
astro-ph.GA 2024-11 conditional novelty 5.0 of 10

A model-dependent orbit superposition reconstruction of APOGEE bulge stars finds the bulge is a 4:3 mix of thick and thin disc populations and is metal-rich relative to the surrounding disc when viewed in 3D.

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