Analytical constraints on generalized <i>β</i> -velocity laws in X-ray-illuminated HMXBs

ABSTRACT We derive analytic constraints on the parameters of the generalized $\beta$-velocity law used to describe radiation-driven winds from OB (O- and B-type) supergiants in wind-fed X-ray binaries. Starting from the radial equation of motion with Castor–Abbott–Klein (CAK) line driving, gas pressure, and X-ray irradiation, we perform an asymptotic analysis of the near-base region and show that a monotonic CAK-type acceleration requires the exponent of the velocity law to satisfy a robust lower bound $\beta _{\min }=1/2$. A separate condition follows from requiring the $\beta$-law to connect smoothly to the subsonic layers: the wind speed at the stellar surface must be at least comparable to the isothermal sound speed, which yields a conservative upper limit on the onset parameter, $b \lesssim 1-\left(a_{\rm s}/v_\infty \right)^{1/\beta }$,and implies $b_{\max } \approx 0.95$–0.99 for typical OB-star parameters. We include the effects of X-ray photoionization and attenuation in an approximate way by introducing an effective force-multiplier slope $\alpha _{\rm eff}$ and an optical-depth parameter controlling the weakening of X-ray inhibition. A scaling analysis shows that strong irradiation favours larger $\beta$ and can lead to the loss of monotonic acceleration, providing an effective system-dependent upper limit on the acceleration index. To assess the consistency of these bounds, we integrate a simplified momentum equation and fit the resulting accelerating solutions with a generalized $\beta$-law. The best-fitting pairs $(b,\beta)$ lie within the analytically allowed domain and occupy the range $\beta \approx 0.6$–1.2, in good agreement with empirical determinations for several persistent high-mass X-ray binaries. Our results provide physically motivated constraints on velocity-law parameters that can guide wind modelling and the interpretation of spectroscopic diagnostics in irradiated massive-star winds.