How Does Viscosity Heat the Disks of Ae Stars?

Classical A-type stars (CAe) are surrounded by gas disks that shine, especially in the Hα line. Traditionally, the heating of these disks was thought to be almost entirely due to photoionization, i.e., the energy supplied by stellar radiation. Recent models, however, have added another ingredient: heat generated by internal friction within the disk, known as viscous heating.

The outcome is striking. For the hottest stars, of type A0‑A1, the effect of viscosity is almost negligible: the disk remains dominated by stellar heating. As we move toward cooler types (A2‑A5), the friction between disk layers becomes more significant, especially if the viscosity parameter, α, is high (≥0.3). Under these conditions the disk warms up, boosting the intensity of Hα emission.

The model also explains why, when observing CAe star populations, the number of emission-line stars drops sharply beyond type A2. If α is sufficiently large, however, even cooler stars (up to A5) could display stronger emissions, a phenomenon still poorly documented because A4‑A5 CAe are rare or understudied.

In short, viscosity is not just a technical detail: it can alter the disk temperature and, consequently, the luminous signature we detect from Earth.

The thermodynamic model adopted for CAe disks relies on Navier‑Stokes equations adapted to a differentially rotating Keplerian fluid. The α parameter, introduced by Shakura and Sunyaev (1973), represents the fraction of total pressure converted into viscous stress; typical values range from 10⁻⁴ to 10⁻¹, but in contexts of strong magnetorotational turbulence (MRI) they can reach 0.3 or higher. Viscous effects manifest as a thermal energy source Q⁺ = (9/4) α P Ω, where P is the local pressure and Ω the angular velocity. In an almost isothermal disk, the increase of Q⁺ raises the equilibrium temperature T_eq, altering the atomic level population distribution and, consequently, the emission of lines such as Hα.

Numerical simulations also include disk radiativity, calculated with Monte‑Carlo methods or 2‑D radiative transfer approaches, to assess the interaction between viscous heating and radiative cooling. A key result is the non‑linear dependence of T_eq on α: for α < 0.1 the temperature increase is marginal, whereas for α ≥ 0.3 a jump of ~2000 K is observed in the inner disk region (r < 5 R_*). This jump is sufficient to shift the Hα formation zone outward, increasing the observed line width. Moreover, viscosity influences the surface density Σ(r) ∝ r^{-p}, with p decreasing as α grows, favoring a flatter material distribution and potentially extending disk lifetime.

Finally, the predicted sharp drop of emission-line stars beyond A2 can be linked to a change in stellar magnetic structure, reducing MRI efficiency and thus the effective α. Future high‑resolution spectroscopy (e.g., ESPRESSO, HARMONI) will test these hypotheses by providing direct α measurements through variable line widths and velocity profiles.