Thermal-timescale accretion does not always yield critical rotation in mass gainers

TL;DR

This work investigates how mass gainers in binaries spin down after rapid thermal-timescale accretion. Using a suite of toy single-star accretion sequences and five detailed binary models, it shows that the surface-to-critical angular velocity $\frac{\omega}{\omega_{\rm crit}}$ declines during thermal relaxation as the star contracts toward thermal equilibrium, with larger decreases when the end-of-accretion disequilibrium is stronger or angular-momentum transport is inefficient. The authors derive a scaling relation linking the post-TE rotation to the degree of thermal disequilibrium via $\frac{\omega_{\rm TE}}{\omega_{\rm crit,TE}} \approx \frac{\omega_{\rm TE}}{\omega_{\rm T}} \left(\frac{R_{\rm TE}}{R_{\rm T}}\right)^{3/2}$, highlighting why spin-down can be substantial. The results imply binary mass transfer does not always yield Be-like critical rotators, instead producing a broad distribution of spin rates applicable to Be-star progenitors, merger remnants, and newly formed massive stars. Observations of sub-critical rotators in post-interaction systems, alongside possible magnetic braking effects in some cases, fit within this framework, underscoring the importance of detailed angular-momentum physics in predicting post-accretion spins.

Abstract

Binary evolution plays a central role in producing rapidly rotating stars. Previous studies have shown that mass gainers in binaries can reach critical rotation after accreting only modest amounts of material, particularly during thermal-timescale Case B mass transfer, where tidal spin-down is ineffective due to wide orbits. However, such rapid accretion often drives the mass gainer out of thermal equilibrium, and its subsequent spin evolution during thermal relaxation has not been analysed in depth. In this study, we construct a suite of accreting detailed single-star models with different accretion prescriptions, which inflate and spin up to critical rotation during the accretion. After the accretion has ended, the models relax thermally and deflate. We find that the ratio of surface to critical angular velocity decreases to subcritical values during thermal contraction, with the magnitude of this decrease correlating with the degree of thermal disequilibrium at the end of accretion. This reduction in fractional critical rotation is even stronger when internal angular momentum transport is inefficient. Detailed binary models show the same trend, indicating that the results from our toy single-star models also apply to real binary evolution. Our results highlight that binary mass transfer does not always produce critically rotating stars, but instead may yield a wide range of spin rates depending on the mass transfer and accretion history. Our findings offer new insights into the rotational properties of mass gainers in binaries, stellar merger products, and newly formed massive stars following accretion.

Figures (18)

• Figure 1: Evolution of a 12$\,\mathrm{M}_\odot$ stellar model from Set I with an accretion rate of $1.0\times 10^{-3}\,\mathrm{M_\odot \,yr^{-1}}$. Accretion begins when the stellar central hydrogen mass fraction drops to 0.5 and ends when the star reaches critical rotation. Panel (a) shows the model’s evolution in the Hertzsprung–Russell diagram. The grey dashed line marks the positions of non-rotating zero-age main-sequence stars, and grey dotted lines show evolutionary tracks of non-rotating single-star models from 2019AA...625A.132S. Open circles denote the four snapshots for which 2D visualizations of the stellar radius and enclosed mass are presented in Fig. \ref{['fig:12Msun_pie']}. Panels (b)–(i) display the time evolution of key physical quantities throughout mass accretion and subsequent thermal relaxation. Thick segments correspond to the accretion phase, and the vertical green line marks the time when thermal equilibrium (TE) is restored. Here we define the restoration of TE as the time when the difference between the nuclear energy generation rate and the stellar luminosity falls below 0.2%. (b) stellar mass; Two key evolutionary times-the end of accretion (1) and the restoration of TE (2) are indicated. (c) stellar luminosity (solid line) and luminosity from nuclear burning (dashed line); To facilitate comparison between the two quantities, the regions beneath the curves are shaded in different colors. (d) moment of inertia $I$ (solid) and the logarithm of the ratio of the total angular momentum $J$ to the surface angular velocity $\omega$ (dashed). (e) stellar radius; Open circles correspond to the last three snapshots shown in Fig. \ref{['fig:12Msun_pie']}. (f) gyration constant $k^2$. (g) stellar surface angular velocity (solid) and critical angular velocity (dashed). (h) ratio of surface angular velocity to the critical value. (i) total angular momentum (solid) and the angular momentum required to maintain critical rotation once thermal equilibrium is restored assuming solid-body rotation (blue asterisk).
• Figure 2: 2D visualization of radius and enclosed mass at four snapshots for a 12$\,\mathrm{M}_\odot$ star from single-star model Set I, with an initial accretion rate of $1.0\times 10^{-3}\,\mathrm{M_\odot\,yr^{-1}}$. The snapshots span evolution phases before mass transfer (MT), during MT, during thermal relaxation (TR), and after restoring thermal equilibrium (TE). The reference time is $t_0=10.06\,$Myr. We ignore deviations from spherical symmetry due to critical rotation and use the averaged radius in the X-Y plane. The color scale denotes enclosed mass, with two distinct colormaps applied below and above $12\,\,\mathrm{M}_\odot$ to differentiate the original stellar mass from the accreted mass. Hatched regions in Panels (b)–(d) highlight the accreted layers. The thin horizontal dashed line marks the stellar radius at $t_0$, and the current stellar radius is labeled in each panel.
• Figure 3: Relationship between stellar spin-down during thermal relaxation and the degree of thermal disequilibrium at the time mass accretion ceases, for all models computed in this study. The spin-down is characterised by $\omega_\mathrm{TE}/\omega_\mathrm{crit,TE}$, where $\omega$ and $\omega_\mathrm{crit}$ denote the surface and critical angular velocities, respectively, and the subscript "TE" refers to the time when thermal equilibrium (TE) is restored. The deviation from TE is quantified by $(R_\mathrm{TE}/R_\mathrm{T})^{3/2}$, where $R_\mathrm{T}$ and $R_\mathrm{TE}$ are the stellar radii at the termination of accretion and at the restoration of TE, respectively. The exponent $3/2$ arises from the dependence of the critical angular velocity on stellar radius (see Appendix \ref{['app_sec:A']} for details). Filled circles without grey edges represent single-star models from Set I, while filled markers with grey edges correspond to single-star models from other sets, with different markers denoting different sets, as indicated in the legend. Open markers represent binary models, with model number labeled beside. The color of each marker (or its edge color for binaries) indicates the mass accretion rate at the termination of accretion. The black dashed line indicates a linear relation with a slope of -1. The grey and cyan shaded regions indicate models that rotate rigidly and differentially, respectively, at the termination of accretion.
• Figure B.1: Internal profiles of a $12\,M_\odot$ single-star model from Set I accreting at $1.0\times10^{-3}\,M_\odot\,\mathrm{yr^{-1}}$, shown as a function of radius coordinate. Colored lines correspond to six snapshots covering the phases before mass transfer, during mass transfer, during thermal relaxation, and after restoring thermal equilibrium. The reference time $t_0=10.06\,$Myr. Panels: (a) enclosed mass; (b) density; (c) luminosity; (d) gravitational energy generation rate (positive=contraction, negative=expansion); (e) opacity; (f) entropy; The thin vertical dashed-dotted lines in Panels (d), (e) and (f) mark the radius of the iron opacity bump. (g) angular velocity; (h) enclosed angular momentum; (i) $\log I$ (solid lines) and $\log(J/\omega_\mathrm{surf})$ (horizontal dotted lines), where $J$, and $\omega_\mathrm{surf}$ denoting the total angular momentum, and surface angular velocity at the corresponding snapshots. The relation $I=J/\omega_\mathrm{surf}$ corresponds to rigid rotation, while $I>J/\omega_\mathrm{surf}$ indicates differential rotation, with the core rotating more slowly than the envelope.
• Figure B.2: Evolution of a 12$\,\mathrm{M}_\odot$ single-star model accreting material at different constant rates. Each line color corresponds to a specific accretion rate, as indicated in the figure legend. Accretion begins when the central hydrogen abundance drops below 0.5 and terminates once the star reaches critical rotation. Panel (a): Evolutionary tracks in the Hertzsprung–Russell diagram. Panels (b)-(h): Time evolution of key stellar properties. (b) stellar mass; (c) moment of inertia $I$ (solid) and the logarithm of the ratio between the total angular momentum $J$ and the surface angular velocity $\omega$ at the termination of mass transfer (dashed); (d) stellar radius; (e) gyration constant $k^2$; (f) surface angular velocity (solid lines) and critical velocity (dashed lines); (g): ratio of surface angular velocity to critical velocity; (h): logarithm of the total angular momentum (solid line) and the total angular momentum required for the corresponding model to rotate at the critical velocities after thermal equilibrium is restored.