[Paper Review] 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

J. Zorec, Y. Frémat+10 more|SPIRE - Sciences Po Institutional REpository|Feb 24, 2017

Stellar, planetary, and galactic studies85 references4 citations

TL;DR

This paper investigates how surface differential rotation in Be stars distorts the interpretation of $V\!\sin i$ measurements and skews the observed distribution of rotational velocity ratios $u = V/V_{\rm c}$. Using Maunder's differential rotation law $\Omega(\theta) = \Omega_0(1 + \alpha\cos^2\theta)$, it shows that negative $\alpha$ values (faster equatorial rotation) lead to underestimation of $V\!\sin i$, while positive $\alpha$ causes overestimation, significantly affecting the inferred number of stars near critical rotation. The study emphasizes that two-dimensional radiation transfer is essential for accurate rotation diagnostics in rapid rotators.

ABSTRACT

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 $u=V/V_ m c$ ($V_ m c$ 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 $u=V/V_ m c$. We conclude that the inferred $V\!\sin i$ is smaller than implied by the actual equatorial linear rotation velocity $V_ m eq$ 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 $V_ m eq\simeq0.9V_ m c$ 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.

Motivation & Objective

• To assess the impact of surface differential rotation on the interpretation of $V\!\sin i$ in Be stars.
• To evaluate how differential rotation alters the observed distribution $\Phi(u)$ of true rotational velocity ratios $u = V/V_{\rm c}$.
• To identify and analyze hidden effects—beyond differential rotation—that obscure the true rotational state from $V\!\sin i$.
• To challenge the assumption of rigid rotation in $V\!\sin i$ analysis and advocate for improved modeling techniques.
• To determine whether the Be phenomenon is primarily driven by critical rotation, given the confounding effects of differential rotation and other line-broadening phenomena.

Proposed method

• Adopted Maunder's differential rotation law $\Omega(\theta) = \Omega_0(1 + \alpha\cos^2\theta)$ to model angular velocity variation with colatitude $\theta$.
• Simulated line broadening under various $\alpha$ values to quantify deviations in measured $V\!\sin i$ from true equatorial velocity $V_{\rm eq}$.
• Used two-dimensional radiation transfer calculations to model gravity darkening and non-uniform surface brightness in rotationally deformed stars.
• Analyzed the effects of macroturbulence, expansion velocities, circumstellar envelopes, and tidal interactions on spectral line profiles.
• Deconvolved rotational broadening functions over wide spectral ranges to detect asymmetries and non-uniform line profiles.
• Evaluated the limitations of classical stellar atmosphere models in handling gravity darkening under differential rotation.

Experimental results

Research questions

• RQ1 How does differential rotation with parameter $\alpha$ affect the measured $V\!\sin i$ relative to the true equatorial velocity $V_{\rm eq}$?
• RQ2 To what extent does differential rotation distort the observed distribution $\Phi(u)$ of $u = V/V_{\rm c}$, especially near critical rotation?
• RQ3 What are the key observational and conceptual uncertainties that compromise the reliability of $V\!\sin i$ as a proxy for true rotation?
• RQ4 Can classical stellar atmosphere models accurately represent gravity darkening in differentially rotating stars with non-uniform $\Omega(\theta)$?
• RQ5 Is the Be phenomenon predominantly linked to critical rotation, or are the observed $V\!\sin i$ distributions misleading due to hidden effects?

Key findings

• For $\alpha < 0$ (faster equatorial rotation), the measured $V\!\sin i$ is systematically smaller than the true $V_{\rm eq}$, leading to underestimation of rotational velocity.
• For $\alpha > 0$ (faster polar rotation), the measured $V\!\sin i$ is larger than $V_{\rm eq}$, causing overestimation of rotation speed.
• Deviations in $V\!\sin i$ are more pronounced for $|\alpha|$ when $\alpha < 0$, indicating stronger distortion of the observed distribution $\Phi(u)$.
• If Be stars have $\alpha < 0$ on average, the number of stars with $V_{\rm eq} \gtrsim 0.9V_{\rm c}$ is larger than expected from $\Phi(u)$; if $\alpha > 0$, the number is smaller.
• Seven additional effects—macroturbulence, expansion velocities, circumstellar envelopes, tidal interactions, gravity darkening non-uniformity, asymmetric broadening functions, and bi-valued $V\!\sin i$ relations—can further obscure rotational information in $V\!\sin i$.
• Two-dimensional radiation transfer in rotationally deformed atmospheres is essential to reliably diagnose rotation and gravity darkening, as classical models fail under differential rotation.