Critical study of the distribution of rotational velocities of Be stars; II: Differential rotation and some hidden effects interfering with the interpretation of the Vsin i parameter
We assume that stars may undergo surface differential rotation to study its impact on the interpretation of $V\!\sin i$ and on the observed distribution $Φ(u)$ of ratios of true rotational velocities $u=V/V_\rm c$ ($V_\rm c$ is the equatorial critical velocity). We discuss some phenomena affecting the formation of spectral lines and their broadening, which can obliterate the information carried by $V\!\sin i$ concerning the actual stellar rotation. We studied the line broadening produced by several differential rotational laws, but adopted Maunder's expression $Ω(θ)=Ω_o(1+α\cos^2θ)$ as an attempt to account for all of these laws with the lowest possible number of free parameters. We studied the effect of the differential rotation parameter $α$ on the measured $V\!\sin i$ parameter and on the distribution $Φ(u)$ of ratios $u=V/V_\rm c$. We conclude that the inferred $V\!\sin i$ is smaller than implied by the actual equatorial linear rotation velocity $V_\rm eq$ if the stars rotate with $α<0$, but is larger if the stars have $α>0$. For a given $|α|$ the deviations of $V\!\sin i$ are larger when $α<0$. If the studied Be stars have on average $α<0$, the number of rotators with $V_\rm eq\simeq0.9V_\rm c$ is larger than expected from the observed distribution $Φ(u)$; if these stars have on average $α>0$, this number is lower than expected. We discuss seven phenomena that contribute either to narrow or broaden spectral lines, which blur the information on the rotation carried by $V\!\sin i$ and, in particular, to decide whether the Be phenomenon mostly rely on the critical rotation. We show that two-dimensional radiation transfer calculations are needed in rapid rotators to diagnose the stellar rotation more reliably.