Tides in Massive Binaries: Numerical Solutions and Semi-Analytical Comparisons

TL;DR

The study systematically benchmarks tidal dissipation in massive binaries by contrasting direct numerical solutions from GYRE-tides with semi-analytic prescriptions used in population-synthesis codes across a broad mass and period range. It shows that before mass transfer, synchronization and orbital-decay timescales predicted by both approaches generally agree within about two orders of magnitude and are often longer than stellar lifetimes, but strong resonances captured by GYRE-tides can yield substantially shorter, system-specific timescales during mass transfer. The PSR J0045–7319 system exemplifies how resonance-driven dynamical tides can reproduce observed orbital decay that semi-analytic methods miss, highlighting the limitations of coarse prescriptions for individual systems. The authors advocate a two-tier approach: use calibrated parameterizations for population-level studies while relying on full numerical tides for interpreting specific binaries and resonant phenomena, with future work to integrate these insights into efficient surrogate models. Overall, the work clarifies the regimes in which linear, structure-averaged tides suffice and where frequency-dependent, resonance-driven tides are essential for accurate predictions and observational interpretation.

Abstract

We present a systematic comparison between the tidal secular evolution timescales predicted by the direct numerical method and those given by the commonly used semi-analytic prescriptions implemented in 1-D hydrostatic binary evolution codes. Our study focuses on binary systems with intermediate- to high-mass primaries ($M_1 = 5$-$50\,M_\odot$), companion masses between $1.4\,M_\odot$ and $10\,M_\odot$, and orbital periods ranging from 0.5 to 50 days. Before mass transfer, both approaches predict synchronization and orbital decay timescales that agree within $\sim$2 orders of magnitude and typically exceed the stellar main sequence lifetime, implying negligible tidal impact on secular orbital evolution. However, the implied dissipation channels differ, and the differences become more pronounced once mass transfer begins. To test the theoretical predictions against observations, we apply both approaches to the well-characterized PSR J0045--7319 system, which has an orbital decay timescale of 0.5 Myr. The numerical solution reveals strong resonances with internal gravity waves, bringing the predicted orbital period change rate close to the observed value. In contrast, the semi-analytic prescriptions predict orbital decay timescales longer than the Hubble time. These results suggest that for population studies, modestly calibrated parameterized equations may suffice, but for individual systems, reliable interpretation requires direct numerical approaches.

Figures (10)

• Figure 1: Comparison of synchronization timescales (top) and orbital decay timescales (bottom) as functions of orbital period for a binary system with $M_1 = 5\,M_\odot$ and $M_2 = 1.4\,M_\odot$, evaluated at the ZAMS where $R_1 = 2.49\,R_\odot$, effective temperature $T_{\rm eff,1}= 17971\, {\rm K}$. The stellar rotation is assumed negligible initially, and the secondary is treated as a point mass. Only the orbital period is varied from 0.5 to 10 days. Results from the full direct solution (GYRE-tides; orange) are compared with those from the semi-analytical prescriptions for convective damping (blue) and radiative damping (pink). The yellow curves correspond to numerical results including only radiative diffusion damping, while the blue dashed lines indicate the semi-analytical prescription with the fast-tide treatment for convective damping. The Hubble time, thermal timescale $t_{\rm kh}$, and nuclear timescale $t_{\rm nuc}$ are shown as black dashed, dot-dashed, and dotted horizontal lines, respectively.
• Figure 2: Tidal evolution diagnostics for a binary system with a $5\,M_\odot$ primary and a $1.4\,M_\odot$ companion, starting at an initial orbital period of 10 days. All panels are plotted against the orbital period $P_{\rm orb}$. Panels (a) and (b): Synchronization timescale $t_{\rm sync}$ and semimajor axis change timescale $t_a$ (both in years), showing MESA radiative damping dominated models (pink), MESA convective damping dominated models (blue), and GYRE-tides results (orange). The Hubble time is indicated by horizontal black lines. Panel (c): Stellar radius $R_1$ (blue), Roche-lobe radius $R_{\rm rl,1}$ (orange dashed), and convective zones (shaded), all in units of $R_\odot$. The orange shaded region corresponds to the most massive convective zone in radial coordinates (1st $R_{\rm conv}$), while the blue shaded region denotes the second most massive convective zone (2nd $R_{\rm conv}$). (d): Ratio of the primary spin period to the orbital period, $P_{\rm rot}/P_{\rm orb}$. The colored points show the evolution computed with radiative (pink) and convective (blue) damping using the semianalytical method. The tidal forcing strength $\varepsilon_\mathrm{T}$ is plotted as a gray dashed line. Diamond symbols mark the initial model.
• Figure 3: Same as Figure \ref{['fig:M1_5_M2_1.4_P_10_secular']}, but with evolutionary time (in $\log\,t$/yr) on the horizontal axis instead of orbital period.
• Figure 4: Tidal secular evolution for a binary system with a $10\,M_\odot$ primary and a $5\,M_\odot$ companion, starting at an initial orbital period of 5 days. The format is the same as in Figure \ref{['fig:M1_5_M2_1.4_P_10_secular']}.
• Figure 5: Same as Figure \ref{['fig:M1_10_M2_5_P_5_secular']}, but with evolutionary time (in $\log\,t$/yr) on the horizontal axis instead of orbital period.