Precessing warped accretion discs in X-ray binaries

We study the radiation‐driven warping of accretion discs in the context of X‐ray binaries. The latest evolutionary equations are adopted, which extend the classical alpha theory to time‐dependent thin discs with non‐linear warps. We also develop accurate, analytical expressions for the tidal torque and the radiation torque, including self‐shadowing. We investigate the possible non‐linear dynamics of the system within the framework of bifurcation theory. First, we re‐examine the stability of an initially flat disc to the Pringle instability. Then we compute directly the branches of non‐linear solutions representing steadily precessing discs. Finally, we determine the stability of the non‐linear solutions. Each problem involves only ordinary differential equations, allowing a rapid, accurate and well‐resolved solution. We find that radiation‐driven warping is probably not a common occurrence in low‐mass X‐ray binaries. We also find that stable, steadily precessing discs exist for a narrow range of parameters close to the stability limit. This could explain why so few systems show clear, repeatable ‘superorbital’ variations. The best examples of such systems, Her X‐1, SS 433 and LMC X‐4, all lie close to the stability limit for a reasonable choice of parameters. Systems far from the stability limit, including Cyg X‐2, Cen X‐3 and SMC X‐1, probably experience quasi‐periodic or chaotic variability as first noticed recently by Wijers and Pringle. We show that radiation‐driven warping provides a coherent and persuasive framework but that it does not provide a generic explanation for the long‐term variabilities in all X‐ray binaries.