When mass transfer feeds its own overflow

Write a small change in how fully the star fits

Remove a little mass from a donor close to its critical surface. Before declaring instability, ask whether its radius relative to the lobe radius increases or decreases. An expanding donor can ease overflow if the lobe expands still faster. The same stellar expansion inside a contracting lobe makes overflow worse.

\zeta_*={d\ln R_d\over d\ln M_d},\qquad\zeta_L={d\ln R_L\over d\ln M_d},\qquad d\ln{R_d\over R_L}=(\zeta_*-\zeta_L)d\ln M_d

Both responses must describe the same removal and angular-momentum prescription. The stellar response also names the adjustment timescale being considered.

Mass loss means $dln M_d<0$ . Thus $zeta_*>zeta_L$ locally reduces overfill, while $zeta_*<zeta_L$ increases it. Pause over this minus sign: a larger response exponent does not mean a larger radius after mass is lost.

Substitute $R_L=af(q)$ and use the preceding chapter’s ledgers.

\zeta_L=2(bq-1)+(1-b)(2\gamma+1){q\over1+q}+(1+bq){d\ln f\over d\ln q}

The first terms describe orbital separation; the last describes lobe shape within that separation. Donor wind and accretor re-emission update the lever-arm factor with the current mass ratio.

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

A star has more than one radius response

Rapid removal leaves little time to rebuild thermal equilibrium. An adiabatic response approximately preserves the entropy distribution of retained layers. Slower removal can instead be compared with thermally adjusted equilibrium structures. These exponents answer different questions about how much adjustment time the star receives.

A core-free, fully convective $n=3/2$ polytrope gives $R_dpropto M_d^{-1/3}$ and therefore $zeta_{ m ad}=-1/3$ : losing mass expands it. This belongs to a specified model, not every giant with a core. Entropy stratification, core mass and evolutionary state change the response.

Another teaching choice uses $zeta_{ m ad}=4$ and $zeta_{ m eq}=0.8$ . If the same lobe response lies between them, rapid loss reduces overfill while thermal adjustment tends to fill the lobe again. Avoiding immediate dynamical growth is different from maintaining long-term thermal equilibrium transfer. The elapsed times require additional stellar and transfer physics.

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

An accurate radius fit can still reverse the sign

The calculation needs how the radius changes with mass ratio, not just the radius itself. A fit with sub-percent radius error does not inherit the same error bound for its derivative. Near equal stellar and lobe responses, a small derivative bias can reverse the feedback sign.

For conservative transfer with the core-free convective response, directly differentiating the critical three-dimensional volume puts local neutrality near $q=0.630686$ . Differentiating Eggleton’s fit instead gives about $0.633980$ . Both are conditional local-model thresholds, not universal critical mass ratios for binaries.

Choose $q=0.6323329$ between them, and remove one ten-thousandth of the initial donor mass. An independent rotated-angle finite-volume calculation increases relative overfill by about $3.45 imes10^{-7}$ . Using the radius fit for the same removal decreases it by about $3.67 imes10^{-7}$ . Keeping the failed fit makes the derivative issue visible; many successful examples far from neutrality would miss it.

A second retained example at equal masses with a stipulated $zeta_*=0.460$ also reverses the sign. The interactive model therefore directly integrates and refines the volume derivative at each selected mass ratio. Numerical refinement and fit bias remain separate. Within their near-threshold envelope it reports no classification. The evidence consists of finite floating-point comparisons. A formal continuum error certificate would require a separate proof.

Reading: Eggleton (1983), Approximations to the radii of Roche lobes · Soberman, Phinney & van den Heuvel (1997), astro-ph/9703016v1

Build a gravitational ledger from a native giant

Moving from local feedback toward an envelope problem requires a stellar structure. A native profile supplies radius as a function of enclosed mass, and a separate definition chooses the core boundary. For now include only the donor’s own gravitational envelope term, written as a positive cost.

E_g=\int_{M_c}^{M_d}{Gm\over r(m)}\,dm={GM_d(M_d-M_c)\over\lambda_gR_d}

Lambda rewrites this profile’s gravity integral under this core definition. It is not a universal giant-star constant. Internal/recombination energy and the companion-envelope potential term remain absent.

Take the named four-solar-mass MESA school snapshot. Select the last original zone below hydrogen mass fraction $0.01$ . Zone 2222 actually has $0.0092680$ , giving $M_csimeq0.482218M_odot$ . Adjacent zone 2221 has $0.0100294$ , with a mass spacing of about $1.97 imes10^{-4}M_odot$ . This is a discrete boundary choice. A small integration residual does not locate the physical remnant boundary to that precision.

The cut gives $R_csimeq0.07614R_odot$ , $E_gsimeq2.776 imes10^{48},mathrm{erg}$ and $lambda_gsimeq0.4545$ . Changing the hydrogen threshold changes the cost. The separate fifteen-solar-mass school profile can be treated similarly, but its evolutionary state differs: the contrast is not an isolated mass trend. Neither profile supplies the internal-energy columns needed here.

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

An energy endpoint must pass two geometrical tests

Choose a $1.4M_odot$ compact companion, let the original giant fill its lobe, and assign a fraction $alpha_{ m CE}$ of orbital-energy release to the gravity cost. The initial orbit contains the full donor mass; the final one contains its core.

E_g=\alpha_{\rm CE}\left({GM_cM_a\over2a_f}-{GM_dM_a\over2a_i}\right),\qquad a_f=\left[{M_d\over M_ca_i}+{2E_g\over\alpha_{\rm CE}GM_cM_a}\right]^{-1}

The bracket is a positive orbital-energy release. Its unassigned fraction is not automatically radiation or ejecta kinetic energy.

Computed conditional final separation versus energy fraction, with separate direct-contact and core-Roche-fit boundaries — Own adaptation from the attributed four-solar-mass native profile. Red dotted: supplied-radius contact. Gray dashed: core Roche-fit boundary. Different line crossings represent different geometrical failures, not merger probabilities. Thoul & Chidester 2022 · CC BY 4.0

Direct Roche volumes give $a_isimeq89.24R_odot$ . For fractions one, one half, one quarter and one tenth, the conditional final separations are approximately $0.4423,;0.2258,;0.1141,;0.04593;R_odot$ . These are endpoint solutions of the scalar equation, not samples along a gas evolution.

Assign the compact companion radius $2 imes10^{-5}R_odot$ . The one-tenth endpoint directly overlaps the two supplied surfaces. The half and quarter endpoints do not overlap, yet the supplied core is outside its own final Roche lobe. At fraction one both supplied radii fit. Keep these three cases distinct: none alone establishes inevitable survival or a computed merger.

The original giant’s cut radius is also not a thermally relaxed remnant radius. Solving pressure, shocks, gas transport and radiation requires a fluid model beyond this scalar equality. Read fluid calculations with those specific missing questions in mind. The near-machine-precision residual of this algebra cannot stand in for the accuracy of a fluid prediction.

Reading: Ivanova et al. (2013), Common envelope evolution; arXiv:1209.4302v2 · Thoul & Chidester (2022), MESA Summer School; Zenodo 7082536

When mass transfer feeds its own overflow

Write a small change in how fully the star fits

A star has more than one radius response

An accurate radius fit can still reverse the sign

Build a gravitational ledger from a native giant

An energy endpoint must pass two geometrical tests

Exercises and explanations

Try the prediction

Keep the threshold, then test the endpoint

Sources and calculation scope