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Sandbox Physics
Binary mass transfer and common envelopes

AC29 · L3

A changing period in Beta Lyrae

Sandbox Physics ·

With historical adopted parameters, Beta Lyrae’s period increase of about 19 seconds per year implies roughly two hundred-thousandths of a solar mass lost annually in a conservative circular model. This tutorial extracts the needed fields from a fixed primary text and recomputes that conditional estimate.

Choose the calculation before collecting numbers

Beta Lyrae is an interacting binary with a complex accretion structure. Its changing period is a useful starting point, but that clock does not come from an illustrative light curve. Harmanec and Scholz combined historical donor radial velocities in 1993, treated instrumental and spectral-line systematics, and derived a quadratic ephemeris. Eclipse timing provided an external comparison.

Our target is the later fixed-parameter calculation in Mennickent and Djurašević 2013 v1: fitted masses from Table 1 and the adopted period trend in section 4. The disc light-curve fit, original radial-velocity fit and evolutionary-track fit are separate analyses. Keeping these layers distinct identifies what this recomputation actually uses.

Download the standalone scientific ZIP from this page and extract it into a new directory. It contains our scientific code, own synthetic checks and model aggregates. The named downloader places original texts, native structures and upstream software files on your computer. Original observation tables remain at the named upstream sources.

Reading: Harmanec & Scholz (1993), A&A279,131–147 · Mennickent & Djurašević (2013), arXiv1303.5812v1

Fetch the fixed text and locate its fields

python verify_package.py
python fetch_sources.py --output inputs
python reproduce.py --inputs inputs --output results
python validate_science.py

Verify package members first, then obtain only the named upstream responses. The downloader checks exact sizes and SHA-256, and decodes only the two selected native profile members. Cache corruption halts source acquisition. The complete 2013 v1 HTML is pinned by this hash, which the scientific parser checks again.

7c218201125132c16ea8265ac1469728561f54d40fad047d57787dd3638741bf

Table 1 gives Md=2.97ModotM_d=2.97M_odot and Ma=13.16ModotM_a=13.16M_odot; section 4 adopts P=12.9,mathrmdP=12.9,mathrm d and dotP=19,mathrms,yr−1dot P=19,mathrm{s,yr^{-1}}. The program locates these fields in the newly obtained text and evaluates the rate from them. Rounded abstract values are not interchangeable with the pinned table values.

The fixed MESA files supply a mathematical mixed-loss reference, while the school profiles support the later gravity integral. Their versions and reuse terms are checked separately. Download availability alone does not establish permission to redistribute old papers or tables. Derived structure materials retain their respective CC BY or CC BY-SA terms.

Reading: Mennickent & Djurašević (2013), arXiv1303.5812v1 · MESA 25.12.1, jdot_ml_check; commit 4a68bfbce0f6930c3f861d0611e914f02569a899 · Thoul & Chidester (2022), MESA Summer School; Zenodo 7082536 · Jared Goldberg (2025), MESA School Leuven Friday model

Convert period growth into a conditional rate

Define dotx>0dot x>0 as mass removed annually from the donor. In a conservative circular model with other orbital torques neglected, total mass and orbital angular momentum remain fixed. Combining the separation response with Kepler’s period gives the following relation.

P˙P=3x˙(1Md−1Ma),x˙=P˙/P3(1/Md−1/Ma){\dot P\over P}=3\dot x\left({1\over M_d}-{1\over M_a}\right),\qquad\dot x={{\dot P}/P\over3(1/M_d-1/M_a)}
The lighter donor makes the bracket positive, so positive period growth corresponds to a positive removal rate under these assumptions.
P˙P=1912.9×86400 yr−1,x˙≃2.17955×10−5M⊙ yr−1{\dot P\over P}={19\over12.9\times86400}\,\mathrm{yr}^{-1},\qquad\dot x\simeq2.17955\times10^{-5}M_\odot\,\mathrm{yr}^{-1}
Convert the period to seconds before dividing by seconds gained each year. Extra digits describe the fixed-input arithmetic, not observational precision.

Inspect the Beta Lyrae output summary to confirm source extraction and the unit conversion, then examine the loss-channel scenarios. Agreement with the paper’s rounded estimate is a reproduction of the adopted calculation. Both estimates share the same published masses and historical trend.

Reading: Mennickent & Djurašević (2013), arXiv1303.5812v1 · Soberman, Phinney & van den Heuvel (1997), astro-ph/9703016v1

Change the loss assumption and retain failures

P˙P=x˙[3Md−3bMa−(1−b)(3γ+1)Md+Ma]{\dot P\over P}=\dot x\left[{3\over M_d}-{3b\over M_a}-{(1-b)(3\gamma+1)\over M_d+M_a}\right]
The accretor retains fraction bb and escaping material carries jmesc=gammaJo/Mj_{ m esc}=gamma J_o/M. Accretor re-emission uses the current mass ratio for gamma; prescribed factors describe other hypotheses.

The outputs vary retention and the loss lever arm. The coefficient in brackets can increase, decrease, or become nonpositive. A nonpositive coefficient does not support positive donor removal for the adopted positive period trend. The program retains those no-positive-rate cases. It neither relabels a negative algebraic answer as positive loss nor removes inconvenient scenarios.

Individual quoted mass errors do not provide the fitted mass covariance, and the selected trend has no accompanying trend uncertainty. These comparisons therefore do not form a complete new posterior. A present-day rate would need new observations, systematic-error treatment and checks of other period-changing mechanisms. The rate here belongs to this historical adopted-parameter calculation.

Reading: Soberman, Phinney & van den Heuvel (1997), astro-ph/9703016v1 · Mennickent & Djurašević (2013), arXiv1303.5812v1

Check the original RV ephemeris by hand

Open the named 1993 scanned PDF at printed p139, equation5. Verify the quadratic coefficient visually before this independent hand calculation: OCR can misread the exponent and squared cycle count. This step explicitly uses coefficients transcribed from that page. The four public commands do not refit the 1532 historical mean radial velocities.

T(E)=T0+P0E+cE2,P0=12.913780 d,c=3.87196×10−6 dT(E)=T_0+P_0E+cE^2,\quad P_0=12.913780\,\mathrm d,\quad c=3.87196\times10^{-6}\,\mathrm d
Cycle number EE is dimensionless and integral; the original ephemeris uses HJD. Its local period is P(E)=P0+2cEP(E)=P_0+2cE, giving dotP=2c/P0dot P=2c/P_0 at the reference cycle, in days per day.
period_days = 12.913780
quadratic_days = 3.87196e-6
seconds_per_julian_year = 2 * quadratic_days / period_days * 86400 * 365.25
print(seconds_per_julian_year)

Using 365.25 days per Julian year gives about 18.924 seconds per year, consistent with the historical rounded value. This and the later adopted 19 seconds are not two unrelated new measurements. This hand calculation retains the original HJD ephemeris and measures a local trend near its reference cycle. Changing time scales or extrapolating arbitrarily far requires a separate treatment.

Reading: Harmanec & Scholz (1993), A&A279,131–147

Extend the replay to native envelope structures

The same workflow decodes mass, radius and hydrogen columns from two exact native ZIP members, chooses each original core-cut zone, and reintegrates gravitational cost. Outputs retain the cut, adjacent zone and hydrogen-threshold sensitivity, then use direct Roche volumes for conditional energy endpoints. This is a fresh profile calculation, not a binding energy reconstructed from a stored lambda. It does not evolve a new MESA star.

Independent checks also cover slow-wind limits, channel-specific ODE ledgers, volume responses and retained radius-fit sign failures. Public Python reintegrates own exported JavaScript synthetic targets with separate coordinates and solvers. It does not regenerate the protected browser Engine. Keep these forms of evidence distinct.

Record the two source roles separately: Beta Lyrae uses historical fitted observational parameters, while the envelope examples use native school structures and chosen energy assumptions. Together they do not measure Beta Lyrae’s common-envelope efficiency or uniquely reconstruct its past from a present binary.

Reading: Thoul & Chidester (2022), MESA Summer School; Zenodo 7082536 · Jared Goldberg (2025), MESA School Leuven Friday model · MESA 25.12.1, jdot_ml_check; commit 4a68bfbce0f6930c3f861d0611e914f02569a899 · Ivanova et al. (2013), Common envelope evolution; arXiv:1209.4302v2

Exercises and explanations

  1. What happens if 19 seconds per year is divided directly by 12.9 without converting the period to seconds?

    Read the explanation

    The fractional period change and the conditional rate become 86400 times too large. Use matching units for the gained time and the period; the year remains in the rate denominator.

  2. A loss scenario has a negative bracket coefficient while the adopted period trend is positive. How should it be reported?

    Read the explanation

    There is no positive donor-removal solution under that scenario. Retain the failed case and state its incompatibility with the adopted trend. Do not turn its negative algebraic rate into positive transfer, or claim that this conditional result uniquely excludes a real physical mechanism.

Try the prediction

Keep the historical estimate and the playable model in their roles

  1. Compare donor wind and accretor re-emission for the same removed amount in the interactive model. The orbital curves can differ.
  2. Use the standalone ZIP to recompute the historical Beta Lyrae rate and loss scenarios. Explain why tracer playback speed cannot supply this rate in physical annual units.
  3. Compare core cuts and energy endpoints of the two native structures. Identify what was freshly read from a real source and what still depends on a chosen physical assumption.

Standalone fixed-source science files

The package runs the source-field extraction, native gravity integrals, conditional endpoints and independent checks. Original responses stay on the reader computer. The scanned 1993 ephemeris is a separate manual reading check; public Python does not refit RVs or regenerate the protected Engine.

Download the complete standalone ZIP

SHA-256: dc8fd19127787fb319ac04dcf2395f72260df7a523c07e663978a35cd4c46f51

Individual files and exact sizes

Sources and calculation scope

  1. Soberman, Phinney & van den Heuvel (1997), astro-ph/9703016v1

    Sections 2–3, angular-momentum loss channels, response exponents and AppendixA. Translate the paper’s donor/accretor names before using its mass-ratio convention.

  2. MESA 25.12.1, jdot_ml_check; commit 4a68bfbce0f6930c3f861d0611e914f02569a899

    The native inlist and run_binary_extras.f90 give an analytic mixed-loss reference. Our ODE checks the mathematics; we did not compile or pass the MESA evolutionary test. Native code is LGPL3-or-later and is not copied into this Lab.

  3. Ivanova et al. (2013), Common envelope evolution; arXiv:1209.4302v2

    Energy formalism, sources/sinks, timescales and core/remnant boundary discussions. The scalar endpoint does not predict the flow or an ejection morphology.

  4. Harmanec & Scholz (1993), A&A279,131–147

    Printed p139, equation 5 and the RV ephemeris discussion. The small quadratic coefficient must be checked on the scanned page. This historical RV analysis is not a new eclipse-time fit.

  5. Mennickent & Djurašević (2013), arXiv1303.5812v1

    Table 1 fitted masses and section 4’s adopted 12.9-day/19-second-per-year calculation. The disc light-curve fit, evolutionary fit and covariance are not reproduced.

  6. Thoul & Chidester (2022), MESA Summer School; Zenodo 7082536

    CC BY 4.0. Exact lab3_4M_solutions/LOGS_cheb_case1/profile1.data from r22.05.1. Our gravity/core-cut aggregates and envelope figure are adaptations with attribution; not a new MESA run or total-age measurement.

  7. Jared Goldberg (2025), MESA School Leuven Friday model

    CC BY-SA 4.0; repository commit ade5bca00e5caae97ac0b6f880f1de7ccead1553, work_minimini3/ALPHA_TEST_LOGS/LOGS_mlt2mesh1time1/profile8.data. Derived structure aggregates retain these terms. This 15-solar-mass model is distinct from the frontier paper’s 12-solar-mass donor.

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