Hurley, Tout & Pols (2002)

Abstract (paraphrased)¶

Presents the Binary Star Evolution (BSE) rapid population-synthesis algorithm, the binary companion to the single-star (SSE) formulae of Hurley, Pols & Tout (2000). On top of single-star evolution it adds the physics of interacting binaries: mass transfer and Roche-lobe overflow, common-envelope evolution, gravitational-radiation and magnetic-braking angular-momentum loss, and — central to its use here — tidal circularisation and synchronisation of eccentric, non-corotating orbits. Using the equilibrium-tide (convective damping) and dynamical-tide (radiative damping) prescriptions, the paper quantifies the systematic effect of tidal friction on synthesised binary populations and shows that orbits generally circularise before Roche-lobe overflow.

The tidal-circularisation timescale (Eqs. 28–29, 41, verified against the PDF)¶

For stars with convective envelopes (equilibrium tide), the circularisation rate is (Eq. 28)

with $q_2 = m/M$ the companion-to-primary mass ratio, $R$ the (primary) stellar radius, $a$ the semi-major axis, and $(k/T)_c$ the apsidal-motion-constant / damping-timescale factor set by the convective eddy turnover (Eq. 30–31, after Rasio et al. 1996). Hurley et al. write it in the equivalent Rasio form (Eq. 29)

For stars with radiative envelopes (dynamical tide, after Zahn 1977) the scaling is steeper in $R/a$ (Eq. 41):

The key physics for progenax is the very strong $(R/a)^8$ dependence (convective case): the circularisation rate plummets with separation, so only short-period orbits are tidally circularised within the relevant lifetimes. This is exactly the schematic $\tau_{\rm circ} \sim (a/R_\star)^8 / [q(1+q)]$ used in the eccentricity chapter — the convective-envelope (equilibrium-tide) form, inverted.

Use in progenax¶

• Eccentricity distributions — the tidal-circularisation timescale (1) motivating the short-period $e \to 0$ behaviour of the period-dependent eccentricity model.

• progenax.binaries — MoeEccentricity reproduces short-period circularisation intrinsically (the Roche ceiling and the $\eta \le -1$ branch drive $e \to 0$); progenax does not integrate the BSE tidal ODEs themselves.

Notes¶

progenax uses Hurley+2002 only for the scaling intuition behind tidal circularisation — the $(R/a)^8$ law that makes circularisation a short-period phenomenon — not as an implemented tidal-evolution integrator. A full BSE-style tidal-evolution treatment (and the stellar-evolution-aware Roche radius needed for evolved-star cutoffs) is a planned startrax coupling. The single-star formulae underlying BSE are Hurley, Pols & Tout (2000).