The Effect of Viscosity on the Temperature of Ae Star Disks

Abstract

This study explores the impact of viscous heating on decretion disks around Classical Ae (CAe) stars, with a focus on modeling the disk’s thermal structure. While photoionization is the dominant heating mechanism, viscous dissipation can play an important role in shaping the disk temperature, particularly for cooler CAe subtypes. Our models incorporate viscosity-driven heating and predict that shear heating has a negligible effect for early A-type stars (A0–A1), but it becomes increasingly significant for later spectral types, especially as the viscosity parameter ($\alpha$) increases. This heating also influences the strength of H$\alpha$ emission. Furthermore, our models predict a sharp decline in the number of emission-line stars beyond spectral type A2, a trend observed in CAe populations. However, for sufficiently high $\alpha$ values (≥0.3), a higher fraction of emission-line objects is expected even among later subtypes, such as A5, despite the lack of well-characterized CAe stars observed beyond the spectral type A4.

1. Introduction

Rapidly rotating OBA-type main-sequence stars can generate equatorial gaseous disks, known as decretion (outflowing) disks, formed from stellar material ejected at the stellar equator. When the expelled stellar material has sufficient angular momentum, it forms a Keplerian disk. These disks are identified by prominent hydrogen emission lines, particularly H$\alpha$, and an infrared excess [1,2]. Stars that display such emission lines in their spectra are known as emission-line stars, with Classical Be (CBe or Be) and Classical Ae (CAe or Ae) stars distinguished by their spectral types. Rapid rotation of the central star, perhaps along with non-radial pulsations (NRPs), episodic outbursts, and binary interactions, contribute to the transport of angular momentum and disk formation, although the precise mechanism(s) remain unclear [3]. Once ejected, the disk material evolves under gravity and viscous forces.

In the Viscous Decretion Disk (VDD) model [4], viscosity plays a crucial role in the outward transport of the angular momentum from the central star, allowing the formation of a stable Keplerian structure. Viscous heating also influences the thermal structure of the disk, affecting emission properties [5]. The long dissipation timescales observed in disk loss events suggest the presence of viscosity in decretion disks, consistent with predictions from viscous diffusion models [6]. Furthermore, H$\alpha$ line profiles and infrared excesses support the viscous transport mechanism, accurately reproducing the observed disk density structure and evolution timescales [7]. Thus, viscosity is a fundamental driver of the dynamics, morphology, and emission of decretion disks. Empirical studies have measured the viscosity parameter ($\alpha$) of [8] in CBe star disks, with values that typically range from $0.1$ to $1.0$. For example, modeling of the Be disk around 28 CMa ($\omega$ CMa) predicts $\alpha \approx 0.2$[9]. Spectroscopic variability has also provided independent constraints on $\alpha$, such as for the Be/X-ray binary X Per, where $\alpha \sim 0.2$ was needed to match emission line oscillations and disk growth timescales [10].

Given the wide range of viscosity values observed in decretion disks, it is clear that viscosity plays a key role in shaping disk dynamics, stability, and thermal structure. A key question is the degree to which viscosity contributes to disk heating and influences the observed emission characteristics. This study investigates these effects by modeling the temperature structure of CAe disks, which are expected to be significantly cooler than those of CBe stars. Specifically, we aim to assess whether radiative heating from the central star alone is sufficient to account for the observed H$\alpha$ emission strength and frequency in CAe stars, particularly in the cooler subtypes (A3/A4), or if additional heating from viscous dissipation is required to reproduce these features. The implications of this question are important: if radiative heating alone proves inadequate, it suggests that viscous heating is not only dynamically relevant but also energetically essential for sustaining observable emission. Finally, while chromospheric activity is known to drive H$\alpha$ emission in late-type stars, it is unlikely to play a significant role in A-type stars, which lack the convective outer layers necessary and dynamic magnetic field to sustain a classical chromosphere. This further strengthens the case for circumstellar disk processes as the dominant emission mechanism in these stars. In this work, we focus on developing the models required for such a comparative analysis, while a detailed confrontation with observational data is beyond the scope of this paper. That comparison, along with a more extensive interpretation of observational trends, is presented in a separate study (Anusha & Sigut 2025, submitted).

This current work summarize the details of our model CAe disks as follows: In Section 2, we provide an overview of the models employed in this study. Section 3 presents the findings from our preliminary analysis, and Section 4 offers a comprehensive discussion of these results.

2. Methods

The thermal structure of circumstellar disks around CAe stars is computed using the Bedisk code [11]. This code ensures radiative equilibrium in a photoionized disk while accounting for heating and cooling processes associated with the nine most abundant elements (H, He, C, N, O, Mg, Si, Ca, and Fe) across multiple ionization stages. A detailed description of the models and an overview of the Bedisk code can be found in [11,12]. The fundamental stellar parameters used for the current analysis in these models are summarized in

Table 1. Adopted stellar parameters. The radii were computed assuming log(g) = 4.0 for all models.

Spectral	${\mathit{T}}_{\mathbf{eff}}$	Mass	Radius
Type	$\left(\mathit{K}\right)$	$\left({\mathit{M}}_{\odot }\right)$	$\left({\mathit{R}}_{\odot }\right)$
A0	9600	2.46	2.60
A1	9200	2.31	2.52
A2	9000	2.21	2.46
A3	8600	2.15	2.43
A4	8400	2.10	2.40
A5	8200	2.04	2.37

. For spectral types ranging from A0 to A5, we adopt mass ($M_{*}$) and effective temperature ($T_{\mathrm{eff}}$) calibrations. The stellar radii ($R_{*}$) are estimated assuming $log\left(g\right)=4.0$.

Each model of the central star is surrounded by an axisymmetric, equatorial circumstellar disk characterized with densities governed by the parameters (${\rho }_0,n,R_d$), defined as

$$\rho (R,Z)\equiv {\rho }_0{\left({\displaystyle \frac{R_{*}}{R}}\right)}^n{e}^{-{(Z/H)}^2},$$

(1)

where $R_{*}\le R\le R_d$. Here, ${\rho }_0$ represents the base disk density, varying within the range [10⁻¹²–2.5 × 10⁻¹⁰] g cm⁻³. The stellar radius $R_{*}$ is taken from Table 1, while $(R,Z)$ denote the cylindrical coordinates spanning the disk. The vertical extent of the disk Z is determined by the function H, defined as

$${\displaystyle \frac{H}{R}}={\displaystyle \frac{c_s}{V_k}}$$

(2)

where $c_s$ is the gas sound speed and $V_k$ is the keplerian rotation velocity of the disk at radius, R. The power-law index n varies between [1.5–4.0], and the outer disk radius $R_d$ spans from 5$R_{*}$ to 50$R_{*}$ based on observations of Be stars [13,14,15]. For every combination of (${\rho }_0,n,R_d$) defining a unique disk density structure, we can compute the temperature structure $T(R,Z)$ using the Bedisk code [11,12].

Including Viscous Shear Heating in the Disk

For CBe stars, the primary source of energy input into the disk is the photoionizing radiation from the central star; however, in A-type stars, the intensity of this radiation field declines rapidly with decreasing $T_{\mathrm{eff}}$, leading to significantly lower disk temperatures (sometimes even below 5000 K, assuming $T_{\mathrm{disk}}$ = 0.6.$T_{\mathrm{eff}}$). To account for this, we include viscous shear heating in the disk temperature calculations. The shear heating rate per unit area at R is

$$D\left(R\right)={\int }_{-\infty }^{+\infty }{\displaystyle \frac{1}{2}}\nu \rho {\left(R{\displaystyle \frac{d\Omega }{dR}}\right)}^{2}dz$$

(3)

Ref. [16] where the integral is over the vertical extent of the disk. Here $\rho$ represents the gas density (Equation (1)), $\left({\displaystyle \frac{d\Omega }{dR}}\right)$ denotes the angular velocity gradient of the disk, and $\nu$ corresponds to the gas viscosity. The viscosity is modeled using the $\alpha$-prescription from [8] where

$$\nu =\alpha c_{s}H,$$

(4)

with $\alpha$ as a dimensionless, constant parameter in the range $[0,1]$. Using the Keplerian rotation of the disk, the $\alpha$ expression for the viscosity, the definition of the disk scale height, H (above), and the sound speed, $c_s^2=(\gamma P/\rho )$, the disk shear heating rate can be expressed as

$$D\left(R\right)={\int }_{-\infty }^{+\infty }\left({\displaystyle \frac{9}{8}}\gamma \alpha P\Omega \right)dz,$$

where P is the gas pressure. We take the integrand to be the shear heating rate per unit volume. Choosing $\gamma =1.33$ (between isothermal and adiabatic), we recover the result of [4],

$$d(R,z)={\displaystyle \frac{3}{2}}\alpha P(R,z)\Omega \left(R\right),$$

(5)

which has been added to the Bedisk code.

For our models, we adopt $\alpha$ values of 0.01, 0.1, 0.3, and 1.0. Consequently, the complete star+disk models in Bedisk are defined by a unique combination of parameters-(${\rho }_0,n,R_d,\alpha$) for each spectral type. This results in 2640 individual models per spectral type, covering a wide range of physical conditions in the circumstellar disk.

3. Results

3.1. Modeling the H$\alpha$ Emission

The broader objective is to determine whether CAe disk models can predict the observed emission seen in H$\alpha$ solely through photoionization or if additional heating mechanisms are required. In this work, we focus on developing and analyzing the disk models, laying the groundwork for such a comparison. To directly test this hypothesis, we compute the H$\alpha$ profiles for every combination of (${\rho }_0,n,R_d,\alpha$) disk parameters and focus only the models that produce detectable emission. Using the Beray code [12], the radiative transfer equation is solved along a large number of parallel rays directed at a distant observer as defined by the inclination angle i (the angle between the stellar rotation axis and the observer’s line-of-sight). An inclination of i = 0° represents a face-on disk, while i = 90° represents an edge-on disk. We have computed H$\alpha$ profiles for 11 different inclination angles in the range 0° ≥ i ≤ 90°. Hence, for each spectral type we have individual H$\alpha$ profile defined by the parameters (${\rho }_0$,n,$R_D$,$\alpha$;i). The number of disk models in each subtype that produces H$\alpha$ emission compared to the total number of disk models defines emission percentage. The relationship between percentage of emission in each spectra type at given levels of viscosity ($\alpha$) is shown in Figure 1.

3.2. The Disk Temperatures

With the model disks, each defined as a combination of ${\rho }_0,n,R_d,\alpha$,i that produce detectable emission now identified, we can compare these models computed using the density-weighted, average disk temperature defined by

$$<T_{\rho }>\equiv {\displaystyle \frac{2\pi }{M_D}}{\int }_{-\infty }^{+\infty }{\int }_{R_{*}}^{R_D}\rho (R,Z)T(R,Z)RdRdZ\mathrm{for}{R}_{*}\le R\le R_d,$$

(6)

Here $M_D$ is the mass of the disk, the gas density of the disk $\rho (R,Z)$ is obtained from Equation (1) and $T(R,Z)$ are the radiative equilibrium temperatures in the disk computed by Bedisk.

The range of density-weighted average disk temperatures computed using Equation (6) for the every combination of disk parameters (${\rho }_0,n,R_d,$i) with detectable H$\alpha$ emission at different levels of viscosity ($\alpha$) is shown in Figure 2. The figure shows a direct relation between viscosity parameter ($\alpha$) and the range of temperatures, with higher viscosity leading to more detectable H$\alpha$ emission and an increase in both the mean disk temperature and the spread of temperature values with emission.

3.3. Observed CAe Stars

In this section, we analyze the number distribution and emission strength of a well-characterized, homogeneous sample of observed CAe stars and compare these trends with those predicted by our models. The analysis is based on the observed variation in the fraction of CAe stars detected across different spectral types, as well as the distribution of H$\alpha$ equivalent widths (EW). The sample of CAe stars was drawn from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST survey [17] Data Release 5 (DR5), which provides low-resolution spectra ($R\approx 1800$). The work of [18] identified 159 CAe stars and independently determined their spectral types, which are used in this analysis. The spectral types of these stars range from A0 to A4, and the frequency distribution of CAe stars across these types is shown in Figure 3. For each star, the H$\alpha$ line was extracted and continuum normalized to measure the equivalent width (EW). The distribution of the measured EWs for each spectral type is presented in Figure 4. Comparing Figure 1 and Figure 3, we observe a steep decline in the number of CAe stars as we move towards cooler subtypes in both our models and the observations. Notably, the disk models (Figure 1) also show that the fraction of emission-line stars for the same spectral type increases with higher values of the viscosity parameter $\alpha$, and they predict a significant number of CAe stars as cool as spectral type A5-which have not been confirmed observationally (Figure 3).

4. Discussion

We aimed to investigate the heating mechanisms of circumstellar disks around main-sequence A-type emission-line (CAe or Ae) stars. In this work, we focus on the theoretical models used to compute the temperature structure of the disks, incorporating both photoionizing radiation from the central star and viscous shear heating, following the $\alpha$-disk prescription of [8]. While we match the trend of emission-line objects across spectral types between the models and observations, a detailed comparison and in-depth analysis of the H$\alpha$ emission strength will be presented in a subsequent work.

Our CAe disk models reveal the following:

• The contribution of shear heating to the predicted disk temperature and fraction of detectable H$\alpha$ emission is negligible for early A-type stars (A0–A1). However, for cooler spectral subtypes, this effect becomes increasingly significant, with the viscosity parameter $\alpha$ influencing the emission fraction (Figure 1). Higher values of $\alpha$ also tend to predict disks with wide range of temperatures having detectable emission, indicating effective heating (Figure 2). This trend suggests that the $\alpha$ parameter could be constrained using observed samples of CAe stars, particularly for later-type A stars.
• The models predict a sharp decline in the number of emission-line stars beyond spectral type A2, a trend that is also evident in observational data (Figure 3). This drop in the predicted number of emission stars is due to the cooler disk temperatures at later spectral types that dramatically reduce the excited state hydrogen level populations. However, for sufficiently high values of $\alpha$ (≥0.3), shear heating raises the disk temperatures, and a greater fraction of emitters is expected even among cooler subtypes, such as A5. Notably, a well-characterized sample of CAe stars beyond spectral type A4 has not been identified.

These results provide insight into the role of viscosity in disk heating and highlight its potential influence on the observed emission properties of CAe stars. Building on the findings of this study, we perform comprehensive comparison between our model predictions and observational data. The results of this extended analysis have been addressed in a separate work that is currently under review.