Accretion in the spin-down regime

Accretion driven millisecond X-ray pulsars can accrete over a wide range of mass flow rates. The pulsations persist even at small accretion rates at which these objects would be expected to be in the propeller stage. We argue that the inner regions of disks around millisecond X-ray pulsars are sufficiently thick that a fraction of the inflowing matter can accrete even in the spin-down regime from regions of the disk away from the disk plane. This allows these systems to be bright throughout a wide range of mass flow rates. We model the lightcurve of SAX J1808.4-3658 during an outburst and show that the rapid decay stage can be modeled with fractional accretion in the spin-down regime.

💡 Research Summary

The paper addresses a long‑standing puzzle in the study of accreting millisecond X‑ray pulsars (MSXPs): why coherent pulsations persist even when the mass inflow rate drops to levels at which the system should enter the propeller (spin‑down) regime and cease accreting. Traditional models define the propeller condition by the magnetospheric radius (rₘ) exceeding the corotation radius (r_c). In that case the centrifugal barrier imposed by the rotating magnetic field is thought to fling incoming disc material away, shutting off accretion and, consequently, pulsations. Observations of sources such as SAX J1808.4‑3658, however, show that pulsations survive deep into the low‑luminosity phase, and the X‑ray light curve exhibits a “rapid decay” stage rather than an abrupt cutoff.

The authors propose that the inner disc of a rapidly rotating neutron star is not a geometrically thin, razor‑thin structure but a vertically thick, pressure‑supported flow. In a thick disc, material at heights above the mid‑plane can follow magnetic field lines that thread the disc at an angle, allowing a fraction of the inflowing gas to slip past the centrifugal barrier and land on the stellar surface even when rₘ > r_c. They formalize this by introducing the fastness parameter ω_* = (rₘ/r_c)^{3/2}. For ω_* > 1 the system is nominally in the spin‑down regime, yet the geometry of a thick disc defines a critical polar angle θ_c = arccos(ω_*^{-1/3}) within which matter can still be channeled onto the star. Integrating over the allowed solid angle yields a fractional accretion efficiency

 f(ω_) = 1 – (3/2)√(1 – ω_^{-2}) + (1/2)(1 – ω_*^{-2})^{3/2}.

This function smoothly declines from f = 1 at ω_* = 1 to f → 0 only asymptotically as ω_* → ∞, meaning that even deep in the propeller regime a non‑zero portion of the disc mass can be accreted.

To test the model, the authors apply it to the well‑studied outburst of SAX J1808.4‑3658. They adopt the standard self‑similar decay of the mass inflow rate in a viscously evolving disc,

 \dot{M}(t) = \dot{M}_c