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REVIEW 3 major objections 5 minor 83 references

Tidal phenomena in the Galactic Center: The curious case of X7

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A grazing collision around 1950 can explain the mysterious dusty source X7 at the Galactic Center.

desk verdict A specific and useful progenitor scenario for X7, but the whole identification is conditional on one contested orbit and the match is partly tuned; worth reviewing, not worth believing yet. read the letter →

arxiv 2504.15337 v2 pith:CAHBYO2S submitted 2025-04-21 astro-ph.GA

Wasif Shaqil , Diego Calderón , Stephan Rosswog , Jorge Cuadra , Anna Ciurlo , Mark R. Morris , Randall D. Campbell , Andrea M. Ghez This is my paper · ORCID
classification astro-ph.GA
keywords GalacticCenterSagittariusA*X7Gobjectstidaldisruptionstellarcollisionstest-particlesimulationsS33/S0-30
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

X7 is a dusty, gas-rich source just 0.02 parsec from Sagittarius A*, already being stretched by the black hole's tides and due to be torn apart near pericenter around 2035. The paper tests three possible origins — a massive star's wind, ejecta from a binary merger, and debris from a grazing stellar collision — and finds that only the collision scenario succeeds. Extrapolating the orbits of 195 stars back in time, the only star that comes close enough to X7 with the right relative velocity and angular momentum is the early B-type star S33/S0-30, whose closest approach was around 1947. Test-particle simulations from that encounter, with an initially elongated 200 AU cloud carrying a velocity gradient and a launch speed near 600 km/s, reproduce X7's orbit, length, orientation, and line-of-sight velocity pattern, leading the authors to conclude that X7 is most plausibly the stripped material from a grazing collision between S33/S0-30 and a stellar-mass black hole or a Jupiter-mass object.

What carries the argument

The central machinery is a set of test-particle simulations of a debris cloud moving only under the point-mass gravity of Sagittarius A* ($4.3\times10^6 M_\odot$), integrated with an adaptive leapfrog scheme. Initial conditions are obtained by extrapolating the observationally constrained orbits of X7 and the 195 stars into the past to locate the closest three-dimensional approach; S33/S0-30 and X7 meet this criterion in 1947 with separation about 610 AU and relative velocity about 505 km/s. The initial velocity of the cloud is fixed by requiring its specific angular momentum to match that of X7's tip, scanning $v_z$ to minimize the 2021 tip separation, and the initial shape that works best is an ellipsoid of length 200 AU at position angle about 75 degrees with a linear velocity gradient of about 100 km/s. This machinery converts the observed orbit into a small set of tunable launch conditions and then tests whether those conditions are physically plausible.

What would settle it

Adopt the Peißker et al. (2024b) orbit for X7, which is anchored by a 1999 L'-band data point: the closest any star in the sample comes to X7 over the past 200 years is more than 4000 AU, so the supposed 1947 encounter and the resulting S33/S0-30 connection disappear entirely. A second decisive test would be to resolve S33/S0-30 with high angular resolution and find no extended dusty envelope or evidence of recent surface agitation.

Watch

Extended reading notes

Core claim

The paper's central claim is that the observed dynamics and structure of X7 can be replicated, to a large extent, by ejecta launched from the star S33/S0-30 in roughly 1950, provided the ejecta were initially elongated (about 200 AU long), carried a velocity gradient of about 100 km/s from tip to tail, and had an initial maximum speed of about 610 km/s. By minimizing the three-dimensional separation and relative velocity between the simulated cloud's tip and X7's observed tip across epochs, the authors obtain a match of about 50 AU in 2021, well within the observational uncertainties. They interpret this as evidence that X7 is the debris of a grazing collision between S33/S0-30 and a field object such as a stellar-mass black hole or a Jupiter-mass object, with the collision happening near the time of closest approach in 1947. They also argue against the two main rival hypotheses: no high-mass-loss star came close enough to X7 to have formed it from a stellar wind, and ejecta from the similarly orbiting source G3 cannot be placed on X7's orbit. The analysis is explicitly conditional on adopting the X7 orbit derived by Ciurlo et al. (2023); if the alternative orbit of Peißker et al. (2024b) is used, no star in the sample has come within 4000 AU of X7 over the past 200 years, and the S33/S0-30 link disappears entirely.

Load-bearing premise

The entire reconstruction rests on the assumption that the Ciurlo et al. (2023) orbit of X7's tip is the correct one; if the alternative orbit derived by Peißker et al. (2024b) is adopted, no star in the 195-star sample comes within 4000 AU of X7 over the last 200 years, and the proposed S33/S0-30 connection vanishes.

Editorial extensions

If this is right

  • If the scenario is correct, X7 will continue to stretch and begin fragmenting as it approaches pericenter around 2035, giving observers a decades-long forecast of a tidal disruption event near Sgr A* and a chance to watch for a resulting change in the black hole's accretion activity.
  • S33/S0-30 should currently show observable aftermath of a grazing collision—an agitated stellar surface or an extended dusty envelope—that can be searched for with high-resolution spectroscopy and imaging.
  • The rarity estimate in the paper implies that only a small fraction of B-type stars in the central parsec should show such filaments at any time; finding additional filaments around other stars would support the collision origin, while finding none would strain it.
  • The paper rules out a dynamical link between X7 and the G object G3, so the two sources should not be treated as a merger pair in future studies of the G-object population.
  • The conclusions depend on the adopted orbit of X7; if the alternative orbit anchored by the 1999 L'-band measurement is confirmed, then the collision scenario for this particular source fails.

Reading between the lines

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

  • Editorial inference: If X7 is truly collision ejecta, then similar elongated dusty streams in the Galactic Center could be fossil records of recent stellar collisions, effectively turning X7-like sources into tracers of the hidden stellar-mass black hole population.
  • Editorial inference: The model's assumed initial conditions could be tested hydrodynamically by simulating a grazing collision itself; a genuine collision should produce a coherent, expanding filament with a particular density and velocity stratification, whereas a wind or merger ejection would not.
  • Editorial inference: A decisive archival test is possible before 2035: re-reducing the 1999 L'-band image that anchors the competing X7 orbit, or obtaining new high-accuracy astrometry of X7's tip, would settle which orbit is correct and therefore whether the S33/S0-30 encounter actually occurred.
  • Editorial inference: If the impactor was a Jupiter-mass object, it may survive as a free-floating planet near Sgr A*, whereas a stellar-mass black hole would leave S33/S0-30 with a modest velocity kick; future astrometry of S33/S0-30's orbit might distinguish the two impactor types.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper investigates the origin of the Galactic Center dusty source X7, which is being tidally stretched by Sgr A* and is predicted to be disrupted around 2035. Under the assumption that X7 is a purely gaseous/dusty cloud with an age shorter than its ~200 yr orbital period, the authors test three formation scenarios: a massive stellar wind/LBV ejection, ejecta from a stellar merger (with G3 as the merger product), and ejecta from a grazing collision of a star with a compact field object. Using a sample of 195 stars with observationally constrained orbits, they find that the star S33/S0-30 had a close approach to X7 (Δr ~ 610 au, Δv ~ 505 km/s) around 1947, and they run test-particle simulations of an initially ellipsoidal cloud launched from S33/S0-30's position. By tuning the launch velocity, cloud length, orientation, and a tip-to-tail velocity gradient, they obtain a simulated cloud whose tip matches X7's 2021 position to ~50 au and whose length, orientation, and line-of-sight velocity structure resemble the observations. They conclude that a grazing collision between S33/S0-30 and a stellar-mass black hole or Jupiter-mass object is a viable origin for X7. They also argue against the wind and merger scenarios and show that stellar-wind ram pressure, a putative SMBH outflow, and a static ISM drag do not markedly alter the cloud's evolution.

Significance. If the central claim holds, X7 would be the first identified debris stream from a stellar grazing collision in the Galactic Center, with a concrete progenitor candidate (S33/S0-30), a launch epoch (~1947), and a falsifiable prediction that S33/S0-30 may still show an agitated state or a dusty envelope. The paper is commendably transparent about the limitations of the analysis: the authors disclose in Appendix B that an alternative published orbit for X7 (Peißker et al. 2024b) removes the S33/S0-30 connection entirely, they report the median reduced chi-square of the fit (χ²_red ~ 8), and they explicitly note that the initial velocity is obtained by solving for the angular momentum of X7 (Eq. 1), so part of the match is built in by construction. The treatment of secondary dynamical effects (WR winds, SMBH outflow, ISM drag) is careful, and the synthesis of Brγ emission for comparison with observations is a strength. However, the significance of the result is currently limited by the unresolved orbit degeneracy for X7 and by the fact that the agreement with the observed orbit is substantially a consequence of the fitting procedure rather than an independent prediction.

major comments (3)
  1. [Appendix B / Sec. 3.1] The entire candidate-selection and the S33/S0-30 connection depend on adopting the Ciurlo et al. (2023) orbit for X7. The authors show in Appendix B and Fig. B.1 that when the Peißker et al. (2024b) orbit is used, no star in the 195-star sample comes within 4000 au of X7 over the past 200 years, i.e., the proposed progenitor disappears entirely. The paper does not provide a quantitative justification for preferring the Ciurlo orbit; it merely notes the discrepancy. Because all subsequent encounter statistics, angular-momentum alignment, and the tuned ejection velocity (Eqs. 2-4) are predicated on this one orbit choice, this is a load-bearing issue. The authors should either re-run the full analysis with the Peißker orbit or provide a decisive, quantitative argument (e.g., astrometric-data quality, epoch coverage, or a model-comparison statistic) for why the Ciurlo orbit is the correct one to use.
  2. [Sec. 3.2.2, Eq. (1)] The initial ejection velocity is not independently predicted but is solved from X7's specific angular momentum using the position of S33/S0-30 in 1947 (Eq. 1), and the cloud length and position angle are then scanned to minimize the tip separation in 2021 (Sec. 3.2.3, Figs. 4 and 5). The resulting agreement of ~50 au in 2021 is therefore partly guaranteed by construction. The paper should state this more explicitly and provide a diagnostic that is not folded into the fitting procedure, such as a prediction of the morphological evolution prior to 2021 or of the pericenter-passage fragmentation pattern, to establish that the model has genuine predictive power beyond reproducing the target orbit.
  3. [Sec. 4.2, Fig. 10] The reported goodness of fit is marginal: the median reduced chi-square over 3000 realizations is ~8, with a minimum of 3.75, even though the model has several tuned parameters (initial velocity magnitude and direction, cloud length, position angle, and velocity-gradient amplitude). For a model with this many adjusted quantities, χ²_red ≈ 8 is not a tight fit, and the text's statement that the model 'can reproduce the observed orbit of X7 to a large extent' (Sec. 4.2) is stronger than the quantitative evidence supports. The authors should discuss this systematically, for example by estimating the effective number of degrees of freedom, or by showing that the residuals are dominated by systematic uncertainties in the orbit of X7 rather than by model deficiency.
minor comments (5)
  1. [Sec. 3, first paragraph] The sentence 'Once found find the best star candidate to be related to X7' contains a grammatical error and should be rephrased.
  2. [Sec. 3.2.3] The description of the ellipsoidal cloud initialization is confusing: the text states that the polar angle 0° < θ < 90° and azimuthal angle 0° < φ < 180° 'have, in practice, no effect on our overall results,' yet the position angle θ_ini = 75° for the ridge is later treated as an important, tuned parameter. Please clarify the distinction between the angular parameters that define the particle distribution within the ellipsoid and the global orientation angle of the ellipsoid's major axis.
  3. [Sec. 5.3] The sentence 'To date, there is no account of such effects for S33/S0-30, but future observations might be able to detect this' is vague; specifying what kind of observational signature (e.g., photometric variability, enhanced mass loss, or a compact companion) would test the collision scenario would strengthen the discussion.
  4. [Appendix B] The fact that the Peißker et al. (2024b) uncertainties are smaller by an order of magnitude is mentioned only in the appendix; this is relevant to the orbit-choice discussion and should at least be noted in Sec. 3.1 where the X7 orbit is adopted.
  5. [Throughout] The notation for the position angle is used inconsistently: θ (polar angle) and θ_ini (initial position angle) appear in close proximity in Sec. 3.2.3, which may confuse readers; distinct symbols (e.g., ψ for the ridge orientation) would help.

Circularity Check

3 steps flagged · score 6.0 of 10

The ejection velocity is defined in Eq. (1) as the velocity giving X7's angular momentum, and the paper itself describes the orbit match as 'by construction'; independent morphology and emission checks keep the circularity partial.

  1. self definitional [Section 3.2.2, Eq. (1).]
    "To find the initial velocity of the cloud, we calculated the velocity required for ejecta from the position (r) of S33/S0-30 in 1947 to achieve the same observed specific angular momentum (l) of the tip of X7. To do so, we solved the following equation to find vX7(t = 1947), lX7 = rS33/S0-30(t = 1947) × vX7(t = 1947)."

    The launch velocity is not an independent prediction: it is constructed as the velocity that puts the ejecta on X7's specific angular momentum. Since the orbit of a test particle around Sgr A* is determined by its initial position and velocity, imposing X7's angular momentum at the launch point builds the target orbit into the initial condition. The later claim that ejecta from S33/S0-30 'can be placed on a similar orbit to X7's' (Section 6, point 3) therefore restates the input constraint rather than testing it.

  2. self definitional [Section 4.1.]
    "The sky-projected orbits have a strong correspondence (by construction) with the observed orbit, the three-dimensional spatial difference of ~50 au (see Table 1), is well within the uncertainty as reported by Ciurlo et al. (2023)."

    This sentence explicitly concedes that the orbital correspondence is by construction. The simulated tip's orbit is forced to match X7's by the choice of launch velocity (same angular momentum) and by minimizing the 2021 tip separation, so the reported ~50 au agreement in 2021 is a restatement of the fitting criterion, not an emergent success of the S33/S0-30 scenario.

1 more flagged steps
  1. fitted input called prediction [Sections 3.2.2-3.2.3 and 4.2.]
    "The agreement is best in the last epoch although this is not surprising since the initial velocity was selected so that tip of the cloud reproduces the position of X7 at this time."

    The 2021 tip position is exactly the quantity minimized in the initial-condition search: |v_ej| is varied to minimize Delta r_tip and Delta v_tip in 2021 (Fig. 4), the initial length of ~200 au is chosen because it minimizes Delta r_tip in 2021 (Fig. 5), and the ridge position angle theta_ini = 75 deg is selected after testing angles. Citing the resulting last-epoch agreement as evidence that the model 'can reproduce the observed dynamics and structure of X7' is circular because the agreement at the fitting epoch is ensured by the minimization. Genuinely independent content remains in epochs and observables not used in the fit, such as the 2002-2015 morphology evolution, tail orientation, line-of-sight velocity crossover, and Br-gamma flux, so the circularity is only partial.

full rationale

The central dynamical match is partially circular. Eq. (1) defines the ejection velocity so that the ejecta carry the same specific angular momentum as the observed X7 tip, and Section 4.1 explicitly calls the orbit correspondence 'by construction'. The 2021 tip agreement is also the minimization target for the launch speed, cloud size, and ridge angle. Those fitted inputs are then presented as the paper's main support for the conclusion that X7 is S33/S0-30 ejecta. What keeps the paper from being fully circular is the set of observables not used in constructing the initial conditions: the simulated cloud's elongation and orientation over 2002-2021, the radial-velocity crossover from tip to tail, and the synthesized Br-gamma flux agree with X7 without being directly fitted (Sections 4.1-4.2, Figs. 8, 9, 11, C.1). The orbital-choice sensitivity shown in Appendix B (no star within 4000 au if the Peissker et al. 2024b orbit is used) is a robustness problem, not a circularity problem, because both orbits are external observational inputs and the paper discloses the alternative. Self-citations to Ciurlo et al. (2023) supply the observed orbit and the grazing-collision idea, but the present argument does not rest on a uniqueness theorem or on unverified claims from those papers. Overall, the dynamical 'prediction' reduces in part to the fit, while the morphological and radiative comparisons retain independent content; score 6.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model's success rests on four free initial-condition parameters, the adopted X7 orbit, the gas-only nature of X7, and an unmodeled collision that is assumed to produce the required ejecta. No new physical entities are introduced.

free parameters (4)
  • initial ejection velocity v_ej = ~610-612 km/s (vx ~ -420 to -426, vy ~ 433-436, vz in -40 to -30 km/s)
    Chosen to match X7's specific angular momentum and to minimize the 2021 tip separation (Sec. 3.2.2, Eq. 1, Fig. 4).
  • initial cloud length L = 200 au
    Scanned over 1-800 au and chosen to minimize the 2021 tip separation (Fig. 5).
  • initial position angle theta_ini = 75 deg
    Tested over 0-90 deg and chosen to reproduce the observed ridge orientation (Sec. 3.2.3).
  • tip-to-tail velocity gradient = ~100 km/s
    Set by hand via coefficient k to spread velocities; no physical derivation is given (Sec. 3.2.3).
assumptions (5)
  • domain assumption X7 is a purely gaseous/dusty source with no internal compact object; therefore its age is less than its ~200 yr orbital period.
    Stated in Sec. 2; if X7 hosted a star, the age constraint and the tidal-disruption picture change.
  • domain assumption The orbit of X7 derived by Ciurlo et al. (2023) is the correct orbit.
    All encounter searches and initial conditions use this orbit; Appendix B shows the Peißker et al. (2024b) orbit eliminates all progenitor candidates (min separation > 4000 au).
  • domain assumption Star orbits are unchanged over the past 200 years, with Sgr A* as the only significant gravitational source.
    Sec. 3.1; two-body relaxation or a collision kick to S33/S0-30 would alter the backward integration.
  • ad hoc to paper A grazing collision can produce the assumed elongated ejecta with a ~100 km/s velocity gradient and ~600 km/s bulk velocity.
    No collision hydrodynamics is modeled; the initial cloud shape is imposed (Sec. 5.3).
  • domain assumption The cloud has negligible self-gravity and internal pressure; test particles trace its motion.
    Supported by the Roche density ratio ~1e-5 (Sec. 2), but drag and pressure are only checked a posteriori.

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Cite this review

Pith. "Pith review of Tidal phenomena in the Galactic Center: The curious case of X7." pith.science (2026). https://pith.science/paper/CAHBYO2S

@misc{pith2026250415337,
  author       = {Pith},
  title        = {Pith review of: Tidal phenomena in the Galactic Center: The curious case of X7},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CAHBYO2S}},
  note         = {Machine review of arXiv:2504.15337}
}
abstract

Several enigmatic dusty sources have been detected in the central parsec of the Galactic Center. Among them is X7, located at only $\sim$0.02 pc from the central super-massive black hole, Sagittarius A* (Sgr A*). Recent observations have shown that it is becoming elongated due to the tidal forces of Sgr A*. X7 is expected to be fully disrupted during its pericenter passage around 2035 which might impact the accretion rate of Sgr A*. However, its origin and nature are still unknown. We investigated the tidal interaction of X7 with Sgr A* in order to constrain its origin. We tested the hypothesis that X7 was produced by one of the observed stars with constrained dynamical properties in the vicinity of Sgr A*. We employed a set of test-particle simulations to reproduce the observed structure and dynamics of X7. The initial conditions of the models were obtained by extrapolating the observationally constrained orbits of X7 and the known stars into the past, making it possible to find the time and source of origin by minimizing the three-dimensional separation and velocity difference between them. Our results show that ejecta from the star S33/S0-30, launched in $\sim$1950, can to a large extent, replicate the observed dynamics and structure of X7, provided that it is initially elongated with a velocity gradient across it, and with an initial maximum speed of $\sim$600~km~s$^{-1}$. Our results show that a grazing collision between the star S33/S0-30 and a field object such as a stellar mass black hole or a Jupiter-mass object is a viable scenario to explain the origin of X7. Nevertheless, such encounters are rare based on the observed stellar dynamics within the central parsec.

Figures

Figures reproduced from arXiv: 2504.15337 by the authors.

Figure 1
Figure 1. Schematic representation of the three hypotheses for the origin of X7 studied in this work. (a) X7 formed from a stellar wind of a star during an episode of high mass loss such as a LBV phase. (b) X7 as the ejecta from a stellar merger via the EKL mechanism, where G3 is the merger product due to similar orbital motion. (c) X7 as the ejecta from a collision of a star with a field object such as a stellar-mass black h… view at source ↗
Figure 2
Figure 2. Three-dimensional separation (left) and relative velocity (right) between stars and X7 as a function of time. The two stars closest to X7, S14/S0-16 and S33/S0-30, are shown. S14/S0-16-X7 is displayed as a dotted blue line and S33/S0-30-X7 as a dashed green line. The calculations are shown in the time periods 1800-2050 (left) and 1900-1950 (right). self-gravity is negligible provided that no compact source is as￾soc… view at source ↗
Figure 3
Figure 3. Hammer projection of the ori￾entation of the angular momentum vec￾tors of the stars with full (orange stars) and incomplete (sky-blue stars) orbital solutions. The vertical dimension repre￾sents the inclination of the orbit (i), and the horizontal dimension represents the longitude of the ascending node (Ω). An edge-on star in a clockwise orbit will be at the top of the hammer projection, and a face-on star will be … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Three-dimensional separation (∆rtip) and relative velocity (∆vtip) between the tip of the simulated cloud and X7 in 2021 as a function of the magnitude of initial velocity (|vej|) of the simulated cloud. The solid blue and dashed orange lines represent ∆rtip and ∆vtip,…
Figure 6
Figure 6. Figure 6: Test-particle simulation of a uniformly sampled ellipsoidal cloud launched from the position of S33/S0-30 in 1947. Each color rep￾resents different times in its dynamical evolution. The sky-projected ob￾served orbit of the tip of X7 is represented as the dashed black l…
Figure 7
Figure 7. Figure 7: Comparison of the sky-projected orbits and position in 2021 of X7 and the tip of the simulated clouds. The dashed black line shows the best-fit orbit of the tip of X7, the light blue shaded region shows the uncertainties with a 68% confidence interval, and the purple c…
Figure 8
Figure 8. Figure 8: Evolution of the inclination and line-of-sight velocity for the best-fitting simulated cloud during the period of the observations. Left panel: Sky projection of the simulated cloud at t = 2003, 2008, 2012, 2016, and 2021 yr. The particles are colored from tip to tail …
Figure 9
Figure 9. Figure 9: Comparison of the morphological evolution of X7 and the simulated cloud. The panels show 1′′× 1 ′′ L ′ -band images highlighting the emission from X7 with contours in the period 2002-2021. The simulated cloud is overlaid with colored markers that encode the line-of-sig…
Figure 10
Figure 10. Figure 10: Histogram of the χ 2 red distribution of the 3000 realizations of the simulated cloud with initial conditions obtained from sampling the posterior distribution of the orbital fit of X7. The solid blue line and orange area represent the results of the fiducial case and…
Figure 11
Figure 11. Figure 11: Comparison of the morphology of X7 and the simulated cloud through the observed (contours) and simulated (color map) Brγ images in 2021. The contour levels represent 0.25 (blue), 0.275 (cyan), and 0.30 (magenta) of the Brγ emission maximum. The sky area shown is 1 ′′×…
Figure 12
Figure 12. Figure 12: Schematic representation of the secondary effects that could affect the dynamical evolution of X7. (a) Ram pressure due to stellar winds pointing toward the center of mass of X7. (b) Ram pressure due to a spherical outflow from Sgr A*. (c) Drag force of the medium (th…
Figure 13
Figure 13. Figure 13: Ratio of magnitude of the acceleration on the simulated cloud due to secondary effects and gravity as a function of time. The effects of the stellar winds of the five closest WR stars, an outflow from Sgr A*, and the drag of the medium are shown as dashed orange, dott…
Figure 14
Figure 14. Figure 14: Comparison of the sky-projected orbits and position in 2021 of X7 and the tip of the simulated clouds from G3 in 1952. The dashed black line shows the best-fit orbit of the tip of X7, the light blue shaded region shows the uncertainties with a 68% confidence interval,…
Figure 15
Figure 15. Figure 15: Density map of the gas structures in the RAMSES simulation for the stellar winds from the WR stars feeding the black hole (Calderón et al. 2025). The gas cells with T < 105 K, inclination −90◦ < i < 0 ◦ , and longitude of ascending node −180◦ < Ω < 0 ◦ are considered …

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Pith tools

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