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The X-ray Variability of the Ultraluminous X-ray Sources in the NGC 4631 galaxy

T0 review · 5 major / 12 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read ULX variability mirrors AGN patterns across mass scales

desk verdict Competent 24-year variability census for five ULXs in NGC 4631; the headline AGN-analogy claim rests on three data points and is fragile. read the letter →

arxiv 2607.06156 v1 pith:DDZ2UICZ submitted 2026-07-07 astro-ph.HE

Jiashun Chen , Jianfeng Wu , Zikun Lin This is my paper
classification astro-ph.HE
keywords ultraluminousX-raysourcesvariabilitynormalizedexcessvariancestructurefunctionactivegalacticnucleiaccretionNGC4631blackholemassscaling
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 paper studies five ultraluminous X-ray sources (ULXs) in the galaxy NGC 4631 using 37 X-ray observations spanning 24 years from Chandra, XMM-Newton, and Swift. The authors construct long-term light curves, model X-ray spectra, and compute two variability statistics — the normalized excess variance (σ²_rms), which measures the fractional amplitude of flux fluctuations, and the structure function (SF²), which measures how variability amplitude grows with time interval. For three of the five sources (X3, X4, X5), they find that σ²_rms decreases with increasing mean X-ray luminosity: brighter sources vary less. For two of those three (X3 and X5), the structure function increases with timescale: flux differences grow over longer baselines. Both trends are well-documented in active galactic nuclei (AGNs), where supermassive black holes accrete gas and produce X-ray variability. The authors argue that if these parallels hold up in larger ULX samples, the same underlying accretion physics may govern X-ray variability across black hole masses ranging from tens of solar masses (ULXs) to millions (AGNs), making ULXs scaled-down laboratories for accretion disk and corona processes. The paper also characterizes each source individually: X1 is consistent with a white dwarf undergoing nuclear burning, X2 is a heavily absorbed source embedded in dense material, X3 shows disk-blackbody spectra with occasional outbursts, X4 is a transient with rapid hour-scale flux changes attributed to an inner Comptonizing region, and X5 is persistently luminous with a power-law spectrum and a possible ~311-day periodic modulation.

What carries the argument

The argument rests on two variability diagnostics: (1) normalized excess variance σ²_rms = [1/(N-1)x̄²] Σ(xᵢ-x̄)² - [1/Nx̄²] Σσ²_err,i, which subtracts measurement noise from total observed variance to isolate intrinsic source variability, normalized by the square of mean luminosity; and (2) the structure function SF²(Δt) = ⟨[log F(t+Δt)/F(t)]²⟩, fitted as a power law SF² = A²·(Δt)^γ, where A is the short-timescale variability amplitude and γ is the slope describing how variability accumulates with time. Together these two statistics, computed from 24 years of multi-instrument X-ray monitoring, allow the authors to place ULXs on the same variability diagnostic plane previously occupied only

What would settle it

If a larger sample of ULXs shows no anti-correlation between σ²_rms and luminosity, or if structure functions do not increase with timescale in the majority of sources, the parallel to AGN variability would dissolve. More immediately, if revised absorption corrections for X4 or recalibrated Swift count rates for X3 change either the σ²_rms or mean luminosity values, the three-point trend could vanish or reverse.

Watch

Extended reading notes

Core claim

The central finding is that three ULXs in NGC 4631 exhibit variability patterns — an anti-correlation between normalized excess variance and luminosity, and a growth of structure function with timescale — that qualitatively mirror established trends in AGN X-ray variability. The key statistical objects carrying this argument are σ²_rms (a dimensionless measure of intrinsic flux variability amplitude) and SF² (a measure of how flux differences scale with the time interval between observations). The anti-correlation between σ²_rms and luminosity means that among these three sources, the ones shining brighter show proportionally smaller fluctuations. The increase of SF² with timescale for X3 (γ

Load-bearing premise

The claim that ULX variability parallels AGN variability rests on three sources, each contributing a single σ²_rms data point, making the anti-correlation a trend across three points rather than a fitted relation with a significance test. The authors acknowledge this limitation, but the abstract and summary present the result more confidently. If even one source's variability or luminosity were revised due to absorption corrections or calibration differences, the qualitative

Editorial extensions

If this is right

  • If the σ²_rms–luminosity anti-correlation is universal across ULXs and AGNs, structure function slopes could serve as a tool to estimate black hole masses in ULXs by analogy with AGN scaling relations, providing mass estimates where dynamical measurements are impossible.
  • The ~311-day possible periodicity in X5, if confirmed by future monitoring, would imply a binary orbital or super-orbital timescale that could constrain the mass ratio and geometry of the system.
  • The distinct structure function behaviors — X3 with steep slope (γ≈0.61) but low short-term amplitude, X4 with high short-term amplitude but flat slope (γ≈0.09) — suggest different accretion geometries or compact object types among ULXs, motivating classification schemes based on variability timescale rather than spectral shape alone.
  • X1's identification as a white dwarf nuclear-burning source, if correct, extends the ULX phenomenon to accretors well below stellar-mass black holes, broadening the mass range over which super-Eddington accretion physics applies.
  • The luminosity–hardness correlation in X5 (harder when brighter) supports super-Eddington funnel beaming models where increased accretion rate narrows the wind funnel and enhances the hard Comptonized component.

Reading between the lines

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

  • The three-point σ²_rms–luminosity trend could be a selection effect rather than a physical relation: with only one data point per source, the trend conflates inter-source differences (different black hole masses, accretion rates, viewing angles) with the intra-source variability–luminosity relation seen in AGNs. A robust test would require measuring σ²_rms at multiple luminosity states within indi
  • If ULXs and AGNs share variability physics, the structure function slope γ should systematically depend on black hole mass, with more massive objects showing shallower slopes (slower variability growth). Testing this would require a heterogeneous sample spanning stellar-mass through intermediate-mass to supermassive black holes with consistent SF analysis.
  • The X4 rapid variability (hours timescale, flat SF slope) resembles behavior seen in neutron star ULXs with accretion column instabilities, raising the possibility that SF slope could discriminate between black hole and neutron star accretors in ULXs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 12 minor

Summary. This manuscript presents a systematic long-term X-ray variability study of five ULXs (X1–X5) in NGC 4631, using 37 archival Chandra, XMM-Newton, and Swift observations spanning 24 years. The authors construct light curves, perform spectral modeling, compute normalized excess variance (σ²_rms), and apply structure function (SF) analysis to characterize variability on timescales from days to decades. They report that σ²_rms is anti-correlated with mean luminosity for three sources (X3, X4, X5) and that SF² increases with timescale for X3 and X5, drawing parallels to AGN variability behavior. The data reduction and spectral fitting follow standard procedures (CIAO, SAS, XSPEC), and the methods are correctly implemented per cited references. The central variability claims, however, rest on a very small number of data points and the manuscript does not adequately address cross-instrument systematics or test the sensitivity of its conclusions to individual observations.

Significance. The paper provides a useful compilation of 24 years of multi-mission X-ray data for the NGC 4631 ULXs and applies standard variability tools (σ²_rms, SF) to a sample that has not been systematically studied in this way before. The individual source descriptions (§5.1) are informative, particularly the discussion of X1's supersoft transient behavior and X4's rapid flux changes. The comparison to AGN scaling relations, while speculative given the sample size, is a legitimate motivating framework. The spectral fitting results are reproducible from the tabulated parameters. However, the significance of the central statistical claims is limited by the three-source sample and the absence of formal correlation tests or sensitivity analyses.

major comments (5)
  1. §4.3, Eq. (2), Table 4: The σ²_rms anti-correlation with luminosity is the paper's central quantitative claim, but it rests on only three data points (one per source: X3, X4, X5), with no formal correlation test (e.g., Pearson or Spearman coefficient, p-value). The claim is presented in the abstract and summary as an established result ('confirming that higher-luminosity ULXs exhibit smaller amplitude of flux fluctuations'), which overstates the evidence. A three-point trend without a significance test does not constitute confirmation. The authors should either downgrade the language to 'suggestive trend' or provide a statistical assessment, and the abstract/summary should be revised accordingly.
  2. §4.3, Table 4 (X3, Swift ObsID 00082263005): X3's σ²_rms is likely dominated by a single Swift observation with only 26 net full-band counts. In this observation, Γ_eff = 1.18±0.52 is derived from a hardness ratio with HR = 0.07±0.21, an uncertainty spanning nearly the full physically plausible range. The resulting luminosity (L_X = 2.5×10³⁹ erg/s) is a factor of 5–8× higher than every Chandra and XMM-Newton measurement of X3 (0.3–0.8×10³⁹; Tables 4, 9). This single point inflates both X3's mean luminosity and its variance. The measurement-error term σ²_err,i in Eq. (2) captures only statistical uncertainty on the luminosity, not the systematic uncertainty from the poorly constrained Γ_eff-to-flux conversion. The paper does not test the sensitivity of σ²_rms to this point—for instance, by fixing Γ_eff to the spectral-fitting value (Γ ≈ 1.0–1.3; Table 9) and recomputing. If this point is,
  3. §4.3, Table 4 (X3, Swift ObsID 00082263005), continued: revised downward, X3's σ²_rms could change substantially, potentially breaking the monotonic three-point anti-correlation. The authors should perform a sensitivity test by excluding or revising this single Swift data point and reporting the effect on the σ²_rms–luminosity trend. This is distinct from a generic small-sample concern: it identifies the specific observation and mechanism through which the central claim is most vulnerable.
  4. §4.4, Eqs. (4)–(6): The structure function is computed by combining luminosities from three instruments (Chandra, XMM-Newton, Swift) with different sensitivities, PSFs, and energy responses. Cross-instrument calibration systematics could introduce artificial scatter or bias the SF²–Δt relation, particularly for X3 and X5 where Swift data with poorly constrained Γ_eff contribute. The paper does not discuss this potential systematic or test whether the SF results are robust when restricted to a single instrument (e.g., Chandra-only). A check on the Chandra-only subset would strengthen the claim that the SF² increase with Δt is intrinsic rather than instrument-dependent.
  5. §4.4, Figures 6–8: The SF power-law slopes are reported as γ ≈ 0.61±0.24 (X3), γ ≈ 0.09±0.05 (X4), and a value for X5 that is not clearly stated in the text. The fit uses an unweighted least-squares method on binned log(SF²) values, but the binning criteria (minimum 10 points per bin, 0.1 dex minimum width) may result in very few bins for sources with sparse data. The number of bins used for each fit, the reduced χ² values, and the residuals are not tabulated. Without this information, the goodness of fit cannot be assessed. Please report the number of bins, χ²_red, and residual statistics for each SF fit.
minor comments (12)
  1. Abstract: 'The structure function values increases' should be 'increase' (subject-verb agreement).
  2. §1, paragraph 3: 'three additional ULXs (X6, X7, and X8) have been identified in NGC 4631 with new XMM-Newton data taken in 2025 (Ducci et al. 2025; Allak et al. 2026)' — the reference to Allak et al. 2026 appears to be a future-dated arXiv preprint; please verify the citation format and availability.
  3. §2: There is a stray period before 'Circular regions centered on the known source coordinates' (beginning of the paragraph describing extraction regions).
  4. Table 3 caption: 'The table lists, for each observation, the mission name, observation ID...' — this caption is repeated for Tables 4–6 via 'The columns have the same definitions as those in Table 3,' but Tables 4–6 include XMM-Newton data with different instrument configurations (MOS1, MOS2, PN) that are not mentioned in the Table 3 caption. Consider adding a note about the XMM-Newton detector identifiers.
  5. Table 9: The flux units are stated as 10⁻¹⁴ erg cm⁻² s⁻¹, whereas Tables 8 and 11 use 10⁻¹³. This inconsistency should be noted or corrected for clarity.
  6. §4.1: 'Source X3 reached the ULX luminosity threshold (10³⁹ erg s⁻¹) in a single Swift observation (ObsID = 00082263005)' — this observation is from 2013, not 2018 as stated later in §5.1.3 ('one Swift observation in 2018'). Please correct the date.
  7. §4.2: The Pearson correlation analysis for X5 excludes the Cycle 1 outlier (ObsID 797) to improve the correlation. The justification (ACIS-S quantum efficiency degradation) is reasonable, but the practice of post-hoc outlier removal before correlation testing should be noted explicitly as a caveat, and the result without removal should also be mentioned.
  8. §5.1.1: 'the thermonuclear burning on the surface of the white dwarf causes the photosphere to expand dramatically to a radius of ~10⁹ cm' — this radius estimate is derived from the blackbody assumption, but the text does not explicitly state the luminosity and temperature used. Please provide the specific values used in the calculation.
  9. §5.2: 'If a fraction of the ULX population are indeed scaled-down analogs of AGNs' — the grammar should be 'is' rather than 'are' (fraction is singular).
  10. Figures 1–8: The figure labels and annotations appear to have rendering issues (unicode escape sequences visible in the text). Please ensure figures are properly rendered in the final version.
  11. §3.1: 'We used PIMMS v4.15 to construct the HR–Γ_eff relation' — the acronym PIMMS should be expanded (Portable, Interactive, Multi-Mission Simulator) at first use, or referenced more formally.
  12. References: Several arXiv preprint references (e.g., Allak et al. 2026, Salvaggio et al. 2022) should be checked for published versions.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for a thorough and constructive report. The referee raises four major points: (1) the σ²_rms–luminosity anti-correlation rests on three data points with no formal correlation test and the language overstates the evidence; (2) X3's σ²_rms may be dominated by a single low-count Swift observation with a poorly constrained Γ_eff, and a sensitivity test is needed; (3) the structure function combines data from three instruments without addressing cross-instrument systematics or testing robustness with a single-instrument subset; and (4) SF fit diagnostics (number of bins, χ²_red, residuals) are not reported. We agree with all four points and will revise the manuscript accordingly. Specifically, we will downgrade the language from 'confirming' to 'suggestive,' add Spearman/Pearson coefficients where applicable, perform the requested sensitivity test on the X3 Swift data point, conduct a Chandra-only SF check, and tabulate all SF fit diagnostics. One partial limitation: for some sources the Chandra-only SF subset is too sparse to yield a meaningful power-law fit, and we will report this transparently.

read point-by-point responses

  1. Referee: §4.3, Eq. (2), Table 4: The σ²_rms anti-correlation with luminosity rests on only three data points (X3, X4, X5) with no formal correlation test. The language in the abstract and summary ('confirming') overstates the evidence.

    Authors: The referee is correct on both counts. With only three data points, no correlation test can yield a statistically meaningful p-value, and the word 'confirming' in the abstract and summary overstates the evidence. We will revise the manuscript as follows. First, we will replace 'confirming' with 'suggesting' or 'indicating a suggestive trend' throughout the abstract, Section 4.3, and the Summary (Section 6). Second, we will add an explicit caveat that N=3 precludes a robust statistical assessment. Third, although we recognize that a formal Spearman or Pearson test on three points carries limited inferential power, we will report the coefficients and p-values for completeness and transparency, so the reader can judge the evidence directly. We will also note in the text that this trend is consistent with the larger-sample results of González-Martín et al. (2011), which provides external support but does not substitute for a larger ULX sample. revision: yes

  2. Referee: §4.3, Table 4 (X3, Swift ObsID 00082263005): X3's σ²_rms is likely dominated by a single Swift observation with only 26 net counts, Γ_eff = 1.18±0.52 from HR = 0.07±0.21, and L_X = 2.5×10³⁹ erg/s, a factor of 5–8× higher than all Chandra/XMM measurements. The measurement-error term captures only statistical uncertainty, not systematic uncertainty from the Γ_eff-to-flux conversion. A sensitivity test excluding or revising this point is needed.

    Authors: This is a well-identified and specific concern. We agree that the Swift observation 00082263005 is the single most influential data point for X3's σ²_rms and that the Γ_eff uncertainty introduces a systematic error not captured by the σ²_err,i term in Equation (2). We will perform the requested sensitivity test in the revised manuscript. Specifically, we will recompute X3's σ²_rms and mean luminosity under two alternative assumptions: (a) excluding the Swift data point entirely, and (b) fixing Γ_eff to the spectral-fitting value (Γ ≈ 1.0–1.3 from Table 9) and recomputing the Swift luminosity. We will report the resulting σ²_rms values and show whether the three-point anti-correlation with X4 and X5 is preserved or broken. We will also add a discussion of the systematic uncertainty from the Γ_eff-to-flux conversion for low-count Swift observations. We note that the referee's comment was truncated in transmission, but the concern about the sensitivity of the σ²_rms–luminosity trend to this single point is clear, and we will address it fully. revision: yes

  3. Referee: §4.4, Eqs. (4)–(6): The structure function combines luminosities from three instruments with different sensitivities, PSFs, and energy responses. Cross-instrument calibration systematics could introduce artificial scatter or bias the SF²–Δt relation. A Chandra-only subset check is needed.

    Authors: We agree that cross-instrument systematics are a legitimate concern when combining Chandra, XMM-Newton, and Swift data in the structure function. We will add a discussion of this potential systematic in Section 4.4. We will also perform the requested Chandra-only SF analysis for X3, X4, and X5. We note that X3 has 11 Chandra detections, X4 has 10, and X5 has 12, which may yield enough pairs for a limited SF analysis, though the time baseline is shorter than the full multi-instrument dataset (Chandra data span 2000–2023, but the 2000 observation is a single epoch and the bulk of Chandra monitoring is from 2022–2023). We will report the Chandra-only SF results where feasible and state explicitly where the data are too sparse for a meaningful power-law fit. For X5, which has the most Chandra detections and the widest Chandra luminosity dynamic range, we expect the most informative single-instrument test. If the Chandra-only SF² still increases with Δt, this will strengthen the claim that the trend is intrinsic. revision: yes

  4. Referee: §4.4, Figures 6–8: SF power-law slopes are reported but the number of bins, reduced χ² values, and residuals are not tabulated. The goodness of fit cannot be assessed. Please report the number of bins, χ²_red, and residual statistics for each SF fit.

    Authors: The referee is correct that these fit diagnostics are essential for assessing goodness of fit and are currently missing from the manuscript. We will add a table (or expand the figure captions) reporting, for each source's SF fit: the number of bins used in the fit, the number of data pairs per bin, the reduced χ², and the residual statistics. We will also state the X5 slope value explicitly in the text, which is currently only shown in the figure. Additionally, we will verify that the binning criteria (minimum 10 points per bin, 0.1 dex minimum width) are clearly described and will report how many bins resulted from the merging procedure for each source. If any fit has very few bins (e.g., 2–3), we will note that the power-law slope is poorly constrained and interpret it accordingly. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: σ²_rms and SF² are computed from observed luminosities via standard formulas; AGN comparison uses external literature.

full rationale

The paper computes σ²_rms (Eq. 2) and SF² (Eqs. 4–6) directly from observed X-ray luminosities using standard formulas from Lanzuisi et al. (2014) and Prokhorenko et al. (2024). No parameter is fitted to a subset of data and then 'predicted' for a closely related quantity. The three-point anti-correlation between σ²_rms and mean luminosity is a direct observation, not a fitted relation repackaged as a prediction. The AGN comparison invokes external literature (Nandra et al. 1997; González-Martín et al. 2011; Prokhorenko et al. 2024) with no author overlap. The spectral model selection follows Soria & Ghosh (2009), but this is a methodological choice re-tested on the expanded dataset, not a self-citation chain that forces the variability results. The derivation chain is self-contained against external benchmarks.

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

The paper introduces no new physical entities, particles, forces, or dimensions. All spectral models (diskbb, powerlaw, bbody, raymond, zedge, phabs, wabs) are standard XSPEC components. The 'fireball' scenario for X1 is attributed to Soria & Ghosh (2009). No new conserved quantities or dimensions are postulated.

free parameters (4)
  • Galactic NH (spectral fitting) = 1.2–1.3 × 10^20 cm^-2
    Fixed to the Galactic column density along the line of sight to NGC 4631, adopted from Soria & Ghosh (2009). Not fitted to the ULX data but set from external measurement.
  • Power-law index Γ_eff (photometry) = varies per observation, ~0.4–2.4
    Derived from hardness ratios via PIMMS interpolation for each observation where soft and hard band counts are available. Used to convert count rates to fluxes. Not a free parameter in the variability analysis itself.
  • SF normalization A = X3: ~0.0975, X4: ~0.369, X5: not stated
    Fitted via least-squares to the binned SF²–Δt relation (Eq. 6). A free parameter of the structure function power-law fit.
  • SF slope γ = X3: 0.61±0.24, X4: 0.09±0.05, X5: not clearly stated
    Fitted via least-squares to the binned SF²–Δt relation (Eq. 6). A free parameter of the structure function power-law fit.
assumptions (4)
  • domain assumption The standard thin-disk spectral model (diskbb) is an adequate approximation for the ULX spectra when photon statistics are sufficient for χ² fitting.
    Used throughout §3.2 for spectral modeling of X2, X3, and X4. The authors note (§5.1.3) that during high-luminosity episodes the standard thin-disk assumption may break down, but it is still used for flux extraction.
  • domain assumption The power-law spectral model with Galactic absorption adequately describes the X5 spectrum across all epochs.
    Used for X5 photometry and spectral fitting (§3.1, §3.2, Table 11). Justified by good χ²/dof values but is a modeling choice.
  • domain assumption The structure function follows a power-law relation with time interval (Eq. 5).
    Assumed in §4.4 following Prokhorenko et al. (2024). The fit quality is assessed via reduced χ² but the power-law form is not tested against alternatives.
  • domain assumption Cross-instrument calibration between Chandra, XMM-Newton, and Swift is sufficiently consistent to combine luminosities in a single light curve.
    The light curves (Fig. 1) combine data from all three missions. The authors use instrument-specific HR–Γ_eff relations and PIMMS for count-rate-to-flux conversion, but no systematic cross-calibration uncertainty is propagated into the variability statistics.

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Cite this review

Pith. "Pith review of The X-ray Variability of the Ultraluminous X-ray Sources in the NGC 4631 galaxy." pith.science (2026). https://pith.science/paper/DDZ2UICZ

@misc{pith2026260706156,
  author       = {Pith},
  title        = {Pith review of: The X-ray Variability of the Ultraluminous X-ray Sources in the NGC 4631 galaxy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DDZ2UICZ}},
  note         = {Machine review of arXiv:2607.06156}
}
read the original abstract

We perform a systematic study on the long-term X-ray variability for the five ultraluminous X-ray sources (ULXs) in the NGC 4631 galaxy (X1-X5), using Chandra, XMM-Newton, and Swift observations covering a 24-year span. Light curves for the five ULXs are presented, while X-ray spectra were modeled for observations with sufficient counts. The normalized excess variance and structure function are utilized to study the X-ray variability behavior of the ULXs. The normalized excess variance is anti-correlated with average X-ray luminosity for three ULXs, indicating that objects with higher average luminosity tend to exhibit relatively lower amplitude of variability. The structure function values increases with time interval in two sources, showing that flux variations become more significant for longer timescales. These trends are also found in the X-ray variability of active galactic nuclei (AGNs). The similarity between ULXs and AGNs, if confirmed for a larger sample of sources, possibly indicates similar underlying physical mechanisms for their X-ray variability.

Figures

Figures reproduced from arXiv: 2607.06156 by the authors.

Figure 1
Figure 1. X-ray light curves of X1–X5 constructed from [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. The zoom-in X-ray light curves of X4. The figure follows the same labeling and annotation conven [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The zoom-in X-ray light curves of X5 in the 21.5–24 year span, also following the same labeling [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The hardness ratio – X-ray luminosity panels for X2–X5. The full-band X-ray luminosities [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: The relation between mean X-ray luminosity in the full-band and the normalized excess variance [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The structure function (SF2 ) of X3. The plot uses logarithmic scales for both axes. The vertical axis shows the values of SF2 , and the horizontal axis shows the time intervals between observations. The blue solid line shows the power-law fitting result for the two pa…
Figure 7
Figure 7. Figure 7: The structure function (SF2 ) of X4. The figure caption is the same as that of [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: The structure function (SF2 ) of X5. The figure caption is the same as that of [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

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Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.