Evolutionary Period Changes for 25 X-ray Binaries and the Measurement of an Empirical Universal Law for Angular Momentum Loss in Accreting Binaries

Bradley E. Schaefer (Louisiana State University)

REVIEW 3 major objections 6 minor 118 references

Evolutionary Period Changes for 25 X-ray Binaries and the Measurement of an Empirical Universal Law for Angular Momentum Loss in Accreting Binaries

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper measures orbital period changes for 25 X-ray binaries and, combined with 52 cataclysmic variables, derives an empirical power-law law for angular momentum loss that it claims is universal across accreting binaries.

desk verdict A valuable Pdot catalog that refutes magnetic braking, but the claimed universal AML law is a same-sample fit whose Mdot exponent is likely contaminated by correlated errors. read the letter →

arxiv 2507.13515 v1 pith:25AQB3LN submitted 2025-07-17 astro-ph.SR astro-ph.HE

Bradley E. Schaefer (Louisiana State University) This is my paper · ORCID

classification astro-ph.SRastro-ph.HE

keywords X-raybinariescataclysmicvariablesorbitalperiodchangeangularmomentumlossmagneticbrakingO-Cdiagramsbinaryevolutionempiricalpowerlaw

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 tries to establish that the slow orbital decay of accreting binaries—both X-ray binaries and cataclysmic variables—is governed by a single empirical angular-momentum-loss law, and that the long-standing magnetic braking model is ruled out by direct measurements. It reports new period-change measures for 25 X-ray binaries from O−C diagrams spanning decades and combines them with 52 cataclysmic variables. After subtracting gravitational radiation and mass-transfer contributions, the residual period change $\dot{P}_{\rm AML}$ is fitted to a power law in orbital period, stellar masses, and accretion rate, yielding one law for periods 0.13–1.0 days and analogous laws for shorter and longer periods. If correct, the three laws together describe the evolution of all 77 systems and give population-synthesis modelers a replacement for the magnetic braking recipe.

What carries the argument

The central tool is the O−C diagram: plots of observed minus calculated eclipse or minimum times fitted with parabolas, whose curvature gives the steady period change $\dot P$. Equations 1–11 then subtract the well-known gravitational-radiation contribution $\dot P_{\rm GR}$ and the mass-transfer contribution $\dot P_{\rm mt}$ from the measured $\dot P$ to isolate the residual $\dot P_{\rm AML}$. Equation 14 converts the resulting 84 measures into a chi-square fit of a power law in $P$, $M_{\rm prim}$, $M_{\rm comp}$, and $\dot M$, producing the fitted exponents in Table 8. The load-bearing step is treating the residual $\dot P_{\rm AML}$ as a real extra angular-momentum-loss signal rather than an artifact of uncertain accretion rates.

What would settle it

Take a system with period near 0.5 day whose masses are measured by eclipses and radial velocities and whose accretion rate is measured independently from X-ray luminosity and a Gaia distance. If its $\dot P$ differs from the Equation 16 prediction by more than the propagated uncertainties, or if a decade-long change in $\dot M$ does not produce the predicted power-law change in $\dot{P}_{\rm AML}$, the universal law would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that the dominant angular momentum loss in accreting binaries is empirically $\dot{P}_{\rm AML} = -1500\times 10^{-12}\, P^{1.29} M_{\rm prim}^{2.75} M_{\rm comp}^{-1.00} \dot{M}_{-8}^{0.43}$ for periods from 0.13 to 1.0 days, with separate fitted power laws for binaries below the period gap and for binaries with $P>1$ day. The same 77 systems show that magnetic braking predictions are wrong by more than an order of magnitude for most systems. The paper concludes that the unknown AML mechanism is controlled by the accretion process, because $\dot{P}_{\rm AML}$ rises with accretion rate, and that the empirical family of laws can stand as a universal description of binary evolution until a physical mechanism is identified.

Load-bearing premise

The fitted law assumes that the accretion rates and wind-capture efficiencies in Table 6 are accurate enough, despite uncertainties of roughly a factor of ten in $\dot M$ and poorly known wind efficiencies, that the residual $\dot{P}_{\rm AML}$ after subtracting GR and mass transfer is a real signal rather than an artifact of those errors.

Editorial extensions

If this is right

The magnetic braking model's single evolutionary track is contradicted: 7 of 8 X-ray binaries with main-sequence companions and most cataclysmic variables deviate from its predictions by orders of magnitude.
Evolution and population-synthesis calculations can replace the magnetic braking recipe with Equations 15–16, which reproduce the observed $\dot{P}_{\rm AML}$ scatter to about 0.33 dex.
Because $\dot{P}_{\rm AML}$ depends on the accretion rate, the dominant loss mechanism must live in the accretion flow, stream, or boundary layer, not in the companion's magnetic wind.
Below the period gap, at least 6 of 18 systems show a non-zero $\dot{P}_{\rm AML}$, so gravitational radiation alone does not drive those binaries.
The period gap and minimum period, long cited as successes of magnetic braking, are no longer evidence for that mechanism.

Reading between the lines

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

If Equation 16 is causal, systems whose accretion rate changes sharply, like the recurrent nova U Sco after its 2010 eruption, should show a corresponding power-law change in $\dot{P}_{\rm AML}$; the paper reports such a jump but leaves it unexplained, so watching the next eruption would directly test the $\dot{M}^{0.43}$ term.
The steep positive exponent on $M_{\rm prim}$ predicts that, at equal period and accretion rate, binaries with more massive white dwarfs or neutron stars lose orbital angular momentum faster; this is a testable ranking within existing eclipsing systems.
If the universal law holds, the minimum period and period gap of cataclysmic variables should be derivable from Equations 15–16 rather than from magnetic braking physics, making the observed period distribution an independent check on the fitted exponents.

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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

Read it for the data, not for the law. Schaefer has put together a genuinely useful catalog of orbital period changes for 25 XRBs, with carefully built O-C diagrams and new timings from TESS, the Harvard plates, and AAVSO. Combining those with 52 CVs and testing the magnetic braking model is legitimate, and the MBM's unique-track claim is convincingly dead for these systems. The GR and mass-transfer subtraction is standard, the worked examples are instructive, and the citation pattern is fine: he engages Knigge et al. 2011, Patterson, and the individual system papers.

The soft spot is exactly where your reader put it. Equation 2 defines Pdot_AML by subtracting a term proportional to Mdot, then Equation 14 regresses that residual on Mdot. With Mdot uncertainties 'typically like a factor of ten' (the paper's own Section 5), correlated errors between the subtracted term and the regressor can produce a positive delta even when the true AML has no Mdot dependence. The same contamination shifts beta and gamma through q. Adding a 0.25-dex systematic to force reduced chi-square near unity does not cure bias in the central value; it only inflates the error bars. So the headline delta ~ 0.43 is not established as a property of nature. The law is also fit to the same 77 systems it claims to describe, with no out-of-sample test. The below-gap fit is especially fragile: huge exponent uncertainties, a reduced chi-square well below unity, and the arbitrary treatment of non-parabolic O-C systems as Pdot=0 with ad hoc uncertainties.

None of this undercuts the Pdot measurements or the MBM failure. But the universal-law claim is too strong. The paper deserves a serious referee. The referee should ask for a synthetic-error test: inject noisy Mdot, show whether the regression recovers delta=0 when the true law has delta=0. An out-of-sample check on a few systems would also help. Without that, the universal law is a fitting result, not a measurement.

For binary evolution modelers and CV/XRB observers, the Pdot table is worth having. I'd bring the paper to reading group and would cite the table; I would not yet cite the law.

Referee Report

3 major / 6 minor

Summary. This paper assembles eclipse and minimum-light timings for 25 X-ray binaries, measures or collects their secular orbital period derivatives, and combines them with 52 cataclysmic variables from the author's prior work. After subtracting gravitational-radiation and mass-transfer contributions (Eqs. 2, 9-11), it forms 84 Pdot_AML values for 77 systems, uses them to test magnetic braking and related AML prescriptions, and finds large discrepancies with the MBM. The paper then fits power laws in P, Mprim, Mcomp, and Mdot to Pdot_AML (Eq. 14), reports three 'universal' empirical AML laws (Eqs. 15-16 and Table 8), and claims these describe the actual evolution of all 77 XRBs and CVs.

Significance. The O-C measurements are a substantial and welcome contribution, especially the long-baseline timing for Sco X-1, Her X-1, and V4641 Sgr, the TESS-based timings, and the careful discussion of systematic jitter in several systems. If the AML-law result survives scrutiny, it would provide an empirically based replacement for the MBM. However, the headline universal law is a least-squares fit to the same systems it claims to describe, and the construction of Pdot_AML from the same Mdot that later serves as a regressor creates a serious correlated-error concern. The paper is therefore significant as a measurement paper and as a falsification test of MBM, but the universal-law claim is not yet established.

major comments (3)

[Sec. 8.2 / Eq. (16), with Sec. 5, Eqs. (2), (10), (14)] The headline Mdot exponent (δ=0.43) may be an artifact of regressing a constructed residual on the same noisy Mdot used to construct it. Pdot_AML is defined by subtracting a mass-transfer term that is linear in Mdot (Eq. 10), and log Mdot is then one of the regressors in Eq. (14). With Mdot uncertainties of 'typically like a factor of ten' as stated in Section 5, an overestimate of Mdot makes the subtracted Pdot_mt too large and drives Pdot_AML more negative, while an underestimate drives it less negative; this built-in anti-correlation can produce a positive δ and can also bias β and γ through q=Mcomp/Mprim even when the true AML has no Mdot dependence. Adding a 0.25-dex systematic error changes the scatter but not the central-value bias, so a reduced chi-square near unity does not validate the law. Please add a synthetic-error test (simulate data with a true Pdot_AML independent of Mdot, add factor-ten Mdot errors, and show that the fitting procedure recovers δ=0) or an errors-in-variables / orthogonal regression treatment.
[Table 8 / Sec. 8.2] The claim of a 'universal' law is not supported by a fit to the same 77 systems from which the law was derived. No out-of-sample or cross-validated test is presented, so the statement that Eqs. (15)-(16) are 'the best representations of the actual evolution for all 77' is a goodness-of-fit restatement, not a predictive test. Additionally, the abstract promises a third law for P>1.0 day, but Table 8 contains only the below-gap fit, the 0.13-1.0 day fit, the HMXB/IMXB fit, and an all-systems fit; long-period CVs such as U Sco, V394 CrA, and T CrB appear only in the 'All' row. The paper either needs to present the dedicated P>1.0-day fit or revise the abstract's claim of three laws.
[Sec. 8.1 / Table 6] The statistical treatment of the fit is under-specified. The text states that a systematic error of 0.25 dex is adopted for all binary groups to bring reduced chi-square near unity, but for the below-gap fit the reduced chi-square is 0.4 after this addition, indicating that the error budget is overestimated rather than calibrated. Table 6 reports the last-column acceptable ranges for Pdot_AML as asymmetric, yet the fitting section does not say how these asymmetric errors are converted into sigma, nor how the factor-ten Mdot uncertainties and the poorly known wind-capture efficiencies epsilon are propagated into the parameter errors quoted in Table 8. Please replace the ad hoc systematic with a transparent likelihood or Monte Carlo propagation, and report the sensitivity of α, β, γ, δ to the assumed Mdot and epsilon errors.

minor comments (6)

[Abstract / Eq. (16)] The units of Pdot_AML, P, masses, and Mdot are only given in the text; the abstract's 'in appropriate units' is too terse for a headline equation.
[Table 6] The 'k' and 'kk' shorthand in the Pdot columns is nonstandard and should be defined in the table caption or replaced by explicit powers of ten.
[Sec. 5, Eq. (11)] The reduction of Eq. (11) to Eq. (10) for epsilon=1 is not shown; a short check would help readers verify that the wind and RLOF cases are consistent.
[Sec. 8.1] The term 'jerks' is introduced for fast O-C kinks; since this is not a standard term, define it at first use.
[Table 8] Chi-square values are quoted without the number of degrees of freedom in the table; the text gives some reduced values, but the table should state the dof explicitly.
[Sec. 7.2] The comparison of 50 measures for 44 systems should state explicitly whether the multiple inter-eruption intervals are treated as independent in the statistical tests.

Circularity Check

1 steps flagged · score 6.0 of 10

The Mdot exponent of the 'universal' AML law is partly self-definitional: Pdot_AML is built by subtracting a term linear in the same Mdot that is later used as a regressor, so the claimed accretion-driven AML is not independently measured.

self definitional [Section 5 (Eqs. 2, 10); Section 7.6; Section 8.2 (Eqs. 14, 16); Table 8]
"The accretion rates are often poorly known, with the real uncertainties typically like a factor of ten. ... ˙PAML = ˙P − ˙PGR − ˙Pmt. (2) ... ˙Pmt = 3P (1 − q) ˙MRLOF Mcomp . (10) ... ˙PAML = −CP αM β prim M γ comp ˙M δ −8. (14) ... That ˙PAML is approximately proportional to ˙M is telling us that the AML mechanism is driven by the physics of the accretion."

For every RLOF system, Eq. 10 makes Pdot_AML = Pdot − Pdot_GR − 3P(1−q)Mdot/Mcomp, so the dependent variable is a decreasing function of Mdot by construction, before any physics. The same Table 6 Mdot then serves as the regressor in Eq. 14; the fitted δ = +0.43 (Eq. 16; +0.87 below the Gap, Eq. 15) is unavoidably shaped by that defining subtraction plus correlated Mdot errors. The paper concedes Mdot is uncertain 'typically like a factor of ten' and wind ϵ is 'only poorly known from theory'; overestimating Mdot pushes Pdot_AML negative, underestimating it pushes Pdot_AML positive, biasing δ positive even if the true extra AML has no Mdot dependence. Reduced chi-square near unity removes scatter, not this central-value bias.

full rationale

The 25 XRB Pdot measures are original O−C timing analyses and the 52 CV measures are published eclipse/pulse timings; these are data, so the paper's self-citations (Schaefer 2023, 2024) are not load-bearing circularity under the hard rules. The MBM and its exponents are attributed to external works (Rappaport et al. 1983; Knigge et al. 2011; Paxton et al. 2015), so the rejection of MBM is a data-vs-model comparison, not a self-citation chain. The universal law (Eqs. 15–16) is explicitly an empirical fit to Eq. 14, so the P and mass exponents are genuine, if noisy, descriptions of the constructed residuals. What is circular is the Mdot dependence: Pdot_AML is defined by subtracting Pdot_mt ∝ Mdot (Eq. 10) from the observed Pdot, and the regression of the result on the same Mdot (Eq. 14) necessarily manufactures a positive δ unless Mdot errors are negligible; the paper states they are a factor of ten. The same construction underlies the below-gap 'proof' of nonzero Pdot_AML (Section 8.1), whose chi-square comparison cannot distinguish a real AML from a biased subtraction. The paper never performs the synthetic-error test that would settle whether δ = 0.43 survives. Some independent content remains (the P^1.29 envelope, the mass exponents, three positive-Pdot_AML outliers), so this is partial, not total, circularity: score 6.

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

The central empirical law rests on multiple fitted constants and exponents, on per-system epsilon values for wind accretion, and on assumptions that Mdot estimates are unbiased enough to isolate Pdot_AML. No new physical entity is introduced; the unknown AML mechanism is a residual, not an independently evidenced object.

free parameters (7)

Prefactor C for the 0.13-1.0 day law = 1500 (+1630/-520) x 10^-12
Least-squares fit to 43 Pdot_AML values, Table 8, Equation 16.
Power-law exponents for the 0.13-1.0 day law = alpha=1.29, beta=2.75, gamma=-1.00, delta=0.43
Fitted to the Pdot_AML data; these are the central empirical constants of the headline claim.
Below-gap prefactor and exponents = C=110, alpha=0.50, beta=-0.3, gamma=-0.5, delta=0.87
Separate fit to 18 below-gap systems, Equation 15, Table 8.
HMXB/IMXB prefactor and exponents = C=10500, alpha=1.43, beta=-0.9, gamma=-0.4, delta=0.1
Separate fit to 13 long-period systems, Table 8.
Ad hoc systematic scatter = 0.25 in log10 Pdot_AML
Added in quadrature to measurement errors so that reduced chi-square is near unity; Section 8.1.
Wind capture efficiency epsilon = 0.0001-0.01, with 0.004-0.00004 for disk cases
Chosen per HMXB from Bondi-Hoyle estimates; directly controls Pdot_AML for wind-fed systems, Section 5.
Jitter errors for individual O-C fits = 0.00027 days for KV UMa, 0.007 days for Her X-1
Added so that individual O-C fits have reduced chi-square near unity, Sections 2.6 and 2.15.

assumptions (6)

standard math The gravitational radiation contribution is exactly Equation 9, the standard Peters formula for point masses.
Used to subtract Pdot_GR from every system, Section 5.
domain assumption Conservative Roche lobe overflow mass transfer with Pdot_mt = 3P(1-q) Mdot_RLOF/Mcomp applies to CVs and LMXBs.
Equation 10 assumes no mass is lost from the binary and no spin-orbit coupling effects.
domain assumption Wind accretion can be represented by a single capture efficiency epsilon in Equation 11.
HMXB Pdot_AML values depend strongly on this poorly constrained efficiency, Section 5.
domain assumption O-C curvature measures steady evolutionary Pdot, with bumps and jerks treated as zero-centered noise.
Used throughout Sections 2 and 3; non-parabolic systems are assigned Pdot=0 with an uncertainty.
domain assumption Accretion-rate estimates are independent of the measured Pdot and accurate enough for subtraction.
The paper warns against using Mdot values based on Pdot models, but many Mdot estimates are order-of-magnitude uncertain.
ad hoc to paper The period-range split into below-gap, 0.13-1.0 day, and longer periods is physically meaningful.
The boundaries are chosen from the data and not independently tested, Section 8.

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Cite this review

Pith. "Pith review of Evolutionary Period Changes for 25 X-ray Binaries and the Measurement of an Empirical Universal Law for Angular Momentum Loss in Accreting Binaries." pith.science (2026). https://pith.science/paper/25AQB3LN

@misc{pith2026250713515,
  author       = {Pith},
  title        = {Pith review of: Evolutionary Period Changes for 25 X-ray Binaries and the Measurement of an Empirical Universal Law for Angular Momentum Loss in Accreting Binaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25AQB3LN}},
  note         = {Machine review of arXiv:2507.13515}
}

abstract

I measure and collect timings of phase markers (like eclipse times) for the orbits of 25 X-ray binaries (XRBs) so as to calculate the steady evolutionary period change ($\dot{P}$). I combine these with my observed $\dot{P}$ measures from 52 cataclysmic variables (CVs). Further, I subtract out the contributions from gravitational radiation ($\dot{P}_{\rm GR}$) and mass transfer ($\dot{P}_{\rm mt}$), deriving the period change from the residual unknown angular momentum loss ($\dot{P}_{\rm AML}$=$\dot{P}$-$\dot{P}_{\rm GR}$-$\dot{P}_{\rm mt}$). I have $\dot{P}_{\rm AML}$ measures for 77 XRBs and CVs, with these being direct measures of the driver of binary evolution. The venerable Magnetic Braking Model (MBM) of binary evolution has its most fundamental predictions tested, with most systems having predictions wrong by over one order-of-magnitude. Other proposed mechanisms to explain the AML also fail, so we are left with no known mechanism that dominates the AML. An alternative path to the AML law is empirical, where my $\dot{P}_{\rm AML}$ measures are fitted to a power-law involving the fundamental binary properties. With this, the dominant AML law for systems with orbital periods ($P$) from 0.13--1.0 days is $\dot{P}_{\rm AML} = -1500\times10^{-12} P^{1.29} M_{\rm prim}^{2.75} M_{\rm comp}^{-1.00}\dot{M}^{0.43}_{-8}$, in appropriate units. Similar AML laws for binaries below the Period Gap and for binaries with $P$$>$1.0 day are derived. These three AML laws are of good accuracy and are the best representations of the actual evolution for all 77 XRBs and CVs of all classes, so the three taken together can be called `universal'.

Figures

Figures reproduced from arXiv: 2507.13515 by the authors.

**Figure 1.** P˙ versus P for the XRBs. The horizontal axis is the logarithm of P in days, with this being the only way to cover the huge range for the binaries. The vertical axis also has to cover a huge range of P˙ , so it must be logarithmic, but with negative and positive values, so I have divided the plot into the upper panel for positive-P˙ and the lower panel for negative-P˙ . This plot is just for the 23 XRBs that have a … view at source ↗

**Figure 2.** O − C curve for XTE J1710-281. This O − C curve is constructed from the 78 times of mid-eclipse using data from five X-ray satellites over the years 1999 to 2017, as given by Jain, Sharma, & Paul (2022). The fiducial ephemeris for O − C uses PO−C=0.1367109674 days and an epoch EO−C=2454410.541569. The data after 2001.0 shows a nice concave-up parabola, with my best-fitting parabola shown as the black curve. The two… view at source ↗

**Figure 3.** TESS folded light curve for Sector 48 for KV UMa (XTE J1118+480). The phase folding is calculated with period of 0.16993394 days and zero phase at BJD 2459623.1474. The 3069 individual fluxes have an RMS scatter (at a given phase) near 0.7 ct/s. The individual points are phase averaged into bins 0.04 wide in phase, as shown by the red squares. With 123 fluxes in each phase bin, the uncertainty in the averages is 0… view at source ↗

Figures from the paper (17 more)

**Figure 5.** Figure 5: TESS curve for Sector 66 for V691 CrA (4U 1822-371). This light curve has been folded, with the phase calculated for a period of 0.2332109571 days and an epoch of 2460110.11180. The 15493 small blue dots are each for one flux measure for a 120 second integration. The r…

**Figure 6.** Figure 6: O − C curve for V691 CrA (4U 1822-371). The top panel is for the 48 X-ray eclipse times from Jain et al. (2010) and Mazzola et al. (2019). The bottom panel shows the 35 optical eclipse times from Bayless et al. (2010) shown as red diamonds, plus the two times from TESS…

**Figure 7.** Figure 7: O − C curve for V2134 Oph (MXB 1658-298). This O − C is calculated with the fiducial ephemeris of PO−C=0.29650453 days and EO−C=2443059.22595. The Xray eclipse times were only observable during outbursts starting in 1976, 1999, and 2015. With only three widely separ…

**Figure 8.** Figure 8: TESS folded light curve for Sector 6 for V616 Mon (A0620-00). These data have 120 second time resolution, spread over a nearly-gap-free 25.8 days centered on 2019.0. The phase folding is with zero phase at BJD 2459215.2884 and period of 0.32301415 days. The 17458 indiv…

**Figure 11.** Figure 11: The Sco X-1 folded light curve for the AAVSO magnitudes from 2005 to 2010. The folding by phase adopts a period of 0.787311 days and has the time of the derived photometric minimum at HJD 2454251.943. The full folded light curve shows the two parallel sinewaves, corre…

**Figure 13.** Figure 13: The Her X-1 folded light curve for TESS Sector 51 from 2022.3. This unfiltered CCD light curve stretches 24.6 days, with few gaps, with 2821 integrations each 600 seconds in duration. The folding by phase adopts a period of 1.70016759 days and the zero phase is at BJD…

**Figure 14.** Figure 14: The Her X-1 folded light curve for the AAVSO magnitudes from 2020.0 to 2022.9. This light curve is composed of 716 CCD Johnson V magnitudes and visual magnitudes. The folding by phase adopts a period of 1.70016759 days and has the time of the derived photometric min…

**Figure 15.** Figure 15: The V4641 Sgr folded light curve for the HCO data from 1895.0–1915.0. The magnitudes were measured multiple times with my by-eye measures and with DASCH, with the averages plotted here. The 20 Johnson B magnitudes (blue diamonds) are sparse, yet nevertheless adequate…

**Figure 18.** Figure 18: O − C curve for V884 Sco. This O − C curve is constructed from the 29 times of mid-eclipse using Xray data from 1972 to 2013, with PO−C=3.411650 days and EO−C=2452176.079. The best-fitting parabola (the thick black curve) is concave-down, so the period is decreasing.…

**Figure 17.** Figure 17: The V884 Sco folded light curve for TESS Sector 39 from 2021.4. This unfiltered CCD light curve covers 24.8 days, with few gaps, with 2417 integrations each 600 seconds in duration. The folding uses a period of 3.41165 days and the zero phase is at BJD 2459376.5484. T…

**Figure 19.** Figure 19: The Cyg X-1 folded light curve for TESS Sector 55 from 2022.6. This unfiltered CCD light curve covers 27.2 days, with few gaps, with 18880 integrations each 120 seconds in duration. The folding uses a period of 5.599848 days and the zero phase is at BJD 2459813.8888.…

**Figure 20.** Figure 20: O − C curve for Cyg X-1. This O − C curve is constructed from the 45 times of conjunction (27 times of primary minima and 18 times when the RV increases past the γ-velocity) from 1939–2022, as given in [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]

**Figure 21.** Figure 21: P˙ versus P for XRBs and CVs. This figure shows the reality of binary evolution. This plot provides the test of model prediction for the driver of the binary evolution. This plot includes the 23 XRBs and the 49 CVs that have parabolic P˙ measures, as a quick pictorial…

**Figure 22.** Figure 22: P˙AML versus P for XRBs and CVs. This figure shows the specific properties that can reveal the real dominant AML mechanism. The format and descriptions for this plot are identical to those in [PITH_FULL_IMAGE:figures/full_fig_p036_22.png]

**Figure 22.** Figure 22: This figure has the same format and legend as [PITH_FULL_IMAGE:figures/full_fig_p037_22.png]

**Figure 23.** Figure 23: O − C curve for U Sco. This updated O − C curve is constructed from seasonal averages of 173 eclipse times from 1989–2024. This plot shows the kinks (i.e., sudden period changes by ∆P) across the four RN eruptions in 1999, 2010, 2016, and 2022. The ∆P changed from ne…

**Figure 24.** Figure 24: Median P˙AML versus M˙ for XRB and CV classes. For the classes and median properties in [PITH_FULL_IMAGE:figures/full_fig_p042_24.png]

Discussion (0). Continue with ORCID to comment.

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