Dynamics of magnetic flux tubes in accretion disks of Herbig Ae/Be stars

Sergey A. Khaibrakhmanov; Alexander E. Dudorov

doi:10.1515/astro-2022-0017

Article Open Access

Dynamics of magnetic flux tubes in accretion disks of Herbig Ae/Be stars

Sergey A. Khaibrakhmanov and Alexander E. Dudorov

Published/Copyright: March 21, 2022

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From the journal Open Astronomy Volume 31 Issue 1

Abstract

The dynamics of magnetic flux tubes (MFTs) in the accretion disk of typical Herbig Ae/Be star (HAeBeS) with fossil large-scale magnetic field is modeled taking into account the buoyant and drag forces, radiative heat exchange with the surrounding gas, and the magnetic field of the disk. The structure of the disk is simulated using our magnetohydrodynamic model, taking into account the heating of the surface layers of the disk with the stellar radiation. The simulations show that MFTs periodically rise from the innermost region of the disk with speeds up to 10–12 km s − 1 . MFTs experience decaying magnetic oscillations under the action of the external magnetic field near the disk’s surface. The oscillation period increases with distance from the star and initial plasma beta of the MFT, ranging from several hours at r = 0.012 au up to several months at r = 1 au . The oscillations are characterized by pulsations of the MFT’s characteristics including its temperature. We argue that the oscillations can produce observed IR-variability of HAeBeSs, which would be more intense than in the case of T Tauri stars, since the disks of HAeBeSs are hotter, denser, and have stronger magnetic field.

Keywords: accretion discs; MHD; ISM: magnetic fields; stars: variables: T Tauri; Herbig Ae/Be

1 Introduction

Accretion disks are commonly observed around young stars. Analysis of contemporary observational data shows that accretion disks of young stars (ADYSs) evolve into protoplanetary disks (PPDs), in which conditions are favorable for planet formation.

Polarization mapping of accretion disks and PPDs shows that they have large-scale magnetic field with complex geometry (Li et al. 2016). Outflows and jets, which are ubiquitous in ADYSs, are indirect signs of the large-scale magnetic field in the system (see review by Frank et al. 2014). Robust measurements of the magnetic field strength in ADYSs are still not possible. There are indications that the magnetic field can be dynamically strong near the inner edge of the disk (Donati et al. 2005). Analysis of the observational constraints on magnetic field strength from measurements of the remnant magnetization of meteorites (Levi 1978) and Zeeman splitting of the CN lines (Vlemmings et al. 2019) shows that the magnetic field strength decreases with distance from the star. The observational data confirm predictions of the theory of fossil magnetic field, according to which the large-scale magnetic field of the ADYSs is the fossil field of the parent protostellar clouds (Dudorov 1995, Dudorov and Khaibrakhmanov 2015).

Magnetohydrodynamic (MHD) modeling of ADYSs has shown that strong toroidal magnetic field is generated in the innermost region of the ADYS, where thermal ionization operates and magnetic field is frozen in gas (Dudorov and Khaibrakhmanov 2014). Runaway generation of the magnetic field in this region can be balanced by magnetic field buoyancy leading to the formation of magnetic flux tubes (MFTs) that float from the disk and carry away excess of its magnetic flux (Khaibrakhmanov and Dudorov 2017). MFTs form in a process of magnetic buoyancy instability (also known as Parker instability, Parker 1979) in the stratified disk with strong planar magnetic field. Formation of MFT has been found both in MHD simulations of solar interior (Vasil and Brummell 2008) and simulations of the accretion disks (Takasao et al. 2018).

Parker instability and rising MFTs can have different manifestations in the accretion disks (see review in Dudorov et al. 2019). Khaibrakhmanov et al. (2018) and Dudorov et al. (2019) have shown that rising MFTs oscillate under certain conditions, and the oscillations can be the source of infrared (IR) variability of accretion disks of T Tauri stars (TTSs). In this work, we further develop approach of Dudorov and Khaibrakhmanov and model the dynamics of the MFT in the accretion disk of typical Herbig Ae/Be star (HAeBeS).

Structure of the article is as follows. In Section 2, we outline the problem statement, describe our model of the dynamics of the MFT as well as the accretion disk model. In Section 3.1, we present results of the simulations of the accretion disk structure. The structure of the disk of the HAeBeS is compared with that of the TTS. Section 3.2 is devoted to the investigation of the dynamics of the MFT in the absence of external magnetic field. Effect of the external magnetic field leading to magnetic oscillations of the MFT is investigated in Section 3.3. We summarize and discuss our results in Section 4.

2 Model

2.1 Problem statement

We consider a toroidal MFT formed inside the accretion disk in the region of effective generation of the magnetic field. The dynamics of unit length MFT is modeled in the slender flux tube approximation. Cylindrical coordinates are adopted, ( r , 0 , z ) , where r is the radial distance from the center of the star, z is the height above the midplane of the disk. The MFT is characterized by radius vector r = ( r , 0 , z ) , velocity vector v = ( 0 , 0 , v ) , cross-section radius a , density ρ , temperature T , and internal magnetic field strength B . The disk has density ρ e , temperature T e , pressure P e , and magnetic field strength B e . The MFT starts its motion at some radial distance r from the star and a height z 0 above the disk’s midplane, z = 0 . The MFT moves in the z -direction under the action of buoyant and drag forces.

2.2 Main equations

We follow Dudorov et al. (2019) and use the system of equations describing the MFT dynamics taking into account the buoyant force, turbulent and aerodynamic drag, radiative heat exchange with the external gas, magnetic pressure of the disk,

(1) d v d t = 1 − ρ e ρ g + f d ,

(2) d r d t = v ,

(3) M l = ρ π a 2 ,

(4) Φ = π a 2 B ,

(5) d Q = d U + P e d V ,

(6) P + B 2 8 π = P e ,

(7) d P e d z = − ρ e g z ,

(8) U = P e ρ ( γ − 1 ) + B 2 8 π ρ ,

where f d is the drag force, M l = const is the mass per unit length of the MFT, Φ = const is the magnetic flux of the MFT, Q is the quantity of heat per unit mass of the MFT, U is the energy of the MFT per unit mass, g z is the vertical component of stellar gravity, and γ is the adiabatic index.

Equations of motion (1) and (2) determine dependences v ( t ) and r ( t ) . Differential equations describing evolution of the MFT’s density and temperature can be deduced by taking time derivative of the energy equation (5) and pressure balance (6) and using the equation of the hydrostatic equilibrium of the disk (7). We define the rate of heat exchange as h c = d Q / d t and estimate it in the diffusion approximation,

(9) h c ≅ − 4 3 κ R ρ 2 σ R T 4 − σ R T e 4 a 2 ,

where κ R is the Rosseland mean opacity adopted from the study by Semenov et al. (2003), σ R is the Stefan-Boltzmann constant.

We introduce non-dimensional variables

(10) u = v / v a , z ˜ = z / H , T ˜ = T / T m , ρ ˜ = ρ / ρ m , t ˜ = t / t A , h ˜ c = h c / h m , a ˜ = a / H , B ˜ = B / B e , g ˜ = g z / f a , f ˜ d = f d / f a , P ˜ = P / ( ρ m v a 2 ) ,

where v a is the Alfvén speed, T m = T e ( z = 0 ) and ρ m = ρ e ( z = 0 ) are the temperature and density in the midplane of the disk, t A = H / v a is the Alfvén crossing time, h m = ε m / t A , ε m is the energy density of magnetic field, and f a = v a / t A . All scales are defined at the midplane of the disk. Then the final equations of the MFT dynamics can be written as (tilde signs are omitted)

(11) d u d t = 1 − ρ e ρ g + f d ,

(12) d z d t = u ,

(13) d T d t = 2 ( γ − 1 ) β × h c β 2 T + C m ρ + ρ e g u C m 2 − P e ρ 3 − γ 2 C m ρ + β 2 T + ( γ − 1 ) P e ρ ,

(14) d ρ d t = − ρ e g u + ( γ − 1 ) h c ρ 3 − γ 2 C m ρ + β 2 T + ( γ − 1 ) P e ρ ,

(15) a = C a ρ − 1 / 2 ,

(16) B = C B ρ ,

where β is the midplane plasma beta, C m = B 0 2 / 4 π ρ 0 2 , C a = a ˜ 0 ρ ˜ 0 1 / 2 , C B = B ˜ 0 / ρ ˜ 0 .

Ordinary differential equations (11)–(14) together with the algebraic equations (15) and (16) form closed system of equations describing the dynamics of the MFT. Eqs. (11)–(14) are supplemented by the initial conditions u ( t = 0 ) = 0 , z ( t = 0 ) = z 0 , T ( t = 0 ) = T e , ρ ( T = 0 ) = ρ 0 , a ( t = 0 ) = a 0 , B ( t = 0 ) = B 0 . Values z 0 and a 0 are the free parameters of the model, while the initial density ρ 0 is calculated from the pressure balance (6) at t = 0 . Initial magnetic field strength B 0 is specified through the initial plasma beta inside the MFT, β 0 , which is also a free parameter.

2.3 Model of the disk

The distributions of the density, temperature, and magnetic field in the disk are calculated using our MHD model of accretion disks (Dudorov and Khaibrakhmanov 2014, Khaibrakhmanov et al. 2017). The disk is considered to be geometrically thin and optically thick with respect to its own radiation. The mass of the disk is small compared to the stellar mass M . Inner radius of the disk is equal to the radius of stellar magnetosphere. Outer radius of the disk is determined as the contact boundary with the external medium.

The model is the generalization of Shakura and Sunyaev (1973) model. In addition to the solution of Shakura and Sunyaev (1973) equations for the low-temperature opacities, we solve the induction equation for magnetic field taking into account Ohmic dissipation, magnetic ambipolar diffusion, magnetic buoyancy, and the Hall effect. The ionization fraction is calculated following Dudorov and Sazonov (1987) taking into account thermal ionization, shock ionization by cosmic rays, X-rays and radionuclides, as well as radiative recombinations and recombinations onto dust grains.

Vertical structure of the disk is determined from the solution of the hydrostatic equilibrium equation (7) for polytropic dependence of the gas pressure on density,

(17) ρ e ( z ) = ρ m 1 − z H k 2 1 k − 1 ,

(18) T e ( z ) = T m 1 − z H k 2 ,

where

(19) H k = 2 k k − 1 H ,

k = 1 + 1 / n , n is the polytropic index, scale height H = v s / Ω k ,

(20) Ω k = G M ⋆ r 3

is the Keplerian angular velocity.

We consider that there is an optically thin hydrostatic corona above the optically thick disk. The corona’s temperature is determined by heating due to absorption of stellar radiation,

(21) T c = 185 f 0.05 L 1 L ⊙ 1 / 4 r 1 au − 1 / 2 K ,

where f is the fraction of the stellar radiation flux intercepted by the disk, L is the stellar luminosity (see Akimkin et al. 2012). Transition from the disk to corona is characterized by an exponential change in temperature over the local scale height H in accordance with the results of detailed modeling of the vertical structure of the accretion disks (see Vorobyov and Pavlyuchenkov 2017).

The model of the disk has two main parameters: turbulence parameter α and mass accretion rate M ˙ .

2.4 Model parameters and solution method

Ordinary differential equations (11)–(14) of the model are solved with the Runge–Kutta scheme of the fourth order with step size control.

Initially, the MFT is in thermal equilibrium with external gas at z 0 = 0.5 H . We performed a set of simulation runs for various initial radii of the MFT a 0 , plasma beta β 0 , and radial distances from the star r . Adopted ranges of the initial parameters are listed in Table 1. Adopted fiducial value of r corresponds to the dust sublimation zone, where gas temperature is 1,500 K.

Table 1

Model parameters: radial distance from the star, initial cross-section radius, and plasma beta of the MFT

Quantity	Range of values	Fiducial value
(1)	(2)	(3)
r	0.012–1 au	0.5 au
a 0	0.01–0.4 H	0.1 H
β 0	0.01–10	1

We consider the accretion disk of HAeBeS with mass 2 M ⊙ , radius 1.67 R ⊙ , luminosity 11.2 L ⊙ , surface magnetic field strength 1 kG, accretion rate M ˙ = 1 0 − 7 M ⊙ / year , and turbulence parameter α = 0.01 . Adopted parameters correspond to the star MWC 480 (Donehew and Brittain 2011, Hubrig et al. 2011). Ionization and magnetic diffusivity parameters are adopted from the fiducial run in Khaibrakhmanov et al. (2017).

3 Results

3.1 Radial structure of the disk

First of all, let us consider the structure of the accretion disk of HAeBeS in comparison with the structure of the disk of typical TTS according to our simulations. Detailed discussion of the structure of TTS disks can be found in our previous papers (Dudorov and Khaibrakhmanov 2014, Khaibrakhmanov et al. 2017).

In Figure 1, we plot the radial profiles of midplane temperature T m , gas surface density Σ , midplane ionization fraction x , and magnetic field strength B z .

$Figure 1 Radial profiles of the midplane temperature (a), surface density (b), midplane ionization fraction (c), and midplane magnetic field strength (d) in the accretion disks of typical TTS (yellow lines) and HAeBeS (blue lines). Empty circle markers show the points at which the modeling of the dynamics of the MFT was performed.$

Figure 1

Radial profiles of the midplane temperature (a), surface density (b), midplane ionization fraction (c), and midplane magnetic field strength (d) in the accretion disks of typical TTS (yellow lines) and HAeBeS (blue lines). Empty circle markers show the points at which the modeling of the dynamics of the MFT was performed.

Figure 1 shows that the structures of the accretion disks of HAeBeS and TTS are qualitatively similar.

Temperature and surface density are decreasing functions of distance, which can be represented as piecewise power law profiles. The local slopes of the T m ( r ) and Σ ( r ) profiles are determined by the parameters of opacity dependence on gas density and temperature (see analysis of the analytical solution of model equations by Dudorov and Khaibrakhmanov (2014)).

The ionization fraction profiles x ( r ) are non-monotonic and have minimum at r min ≈ 0.3 au in the case of TTS and 1 au in the case of HAeBeS. The ionization fraction is higher closer to the star, r < r min , due to thermal ionization of alkali metals and hydrogen. Growth of the x further from the minimum, r > r min , is explained by decrease in gas density and corresponding increase in the intensity of ionizing radiation by external sources. Local peak in the x ( r ) profiles at r ≈ 1 (TTS) and 4 au (HAeBeS) is due to evaporation of icy mantles of dust grains.

Intensity of the vertical component of the magnetic field B z generally decreases with distance. In the region of thermal ionization, r < r min , the magnetic field is frozen into gas and B z ∝ Σ . In the outer region, r > r min magnetic ambipolar diffusion reduces magnetic field strength by 1–2 orders of magnitude as compared to the frozen-in magnetic field. For example, magnetic field strength is 0.1 G near the ionization minimum.

Comparison of the simulation results for HAeBeS and TTS shows that the accretion disk is hotter and denser in the former case at any given r . This is because the disk of HAeBeS has higher accretion rate, which leads to more intensive turbulent heating of the gas in the disk. As a consequence, the size of the innermost region, where runaway growth of the magnetic field is possible due to high ionization level, is more extended in the case of the HAeBeS. For adopted parameters, this region ranges from the inner boundary of the disk, r in = 0.012 au , up to r ≈ r min = 1 au . Magnetic field strength is greater in the case of HAeBeS.

3.2 MFT dynamics without external magnetic field

In this section, we study the dynamics of the MFT in the disk of HAeBeS in the absence of the magnetic field outside the MFT.

In Figure 2, we plot dependences of the MFT’s speed, density, radius, and temperature on the z -coordinate at r = 0.5 au for different initial cross-section radii, a 0 = 0.01 , 0.1, 0.2, and 0.4 H .

$Figure 2 Dynamics of the MFTs of various initial cross-section radii a 0 {a}_{0} in the absence of the external magnetic field. Dependences of the MFT’s speed (panel a), density (b), cross-section radius (c), and temperature (d) on the z z -coordinate are shown. Vertical lines show the surface of the disk. Grey dashed lines in panels (b) and (d) delineate the corresponding profiles of the disk’s density and temperature. Initial parameters of the MFT: r = 0.5 au r=0.5\hspace{0.33em}{\rm{au}} , z 0 = 0.5 H {z}_{0}=0.5\hspace{0.33em}H , and β 0 = 1 {\beta }_{0}=1 .$

Figure 2

Dynamics of the MFTs of various initial cross-section radii a 0 in the absence of the external magnetic field. Dependences of the MFT’s speed (panel a), density (b), cross-section radius (c), and temperature (d) on the z -coordinate are shown. Vertical lines show the surface of the disk. Grey dashed lines in panels (b) and (d) delineate the corresponding profiles of the disk’s density and temperature. Initial parameters of the MFT: r = 0.5 au , z 0 = 0.5 H , and β 0 = 1 .

Figure 2(a) shows that thinner MFT, a 0 = 0.01 H , is characterized by three stages of evolution. First the MFT accelerates inside the disk, then it rapidly decelerates near the surface, z ≈ 2.6 H , and after that it again rises with acceleration in the corona of the disk. Finally, the MFT dissipates in the corona, in a sense that its radius grows fast and becomes comparable with the half-thickness of the disk, as shown in Figure 2(c). Hence, MFTs will form outflowing magnetized corona of the disk, as in the case of TTS discussed by Dudorov et al. (2019). The MFTs of intermediate initial radii, a 0 ∼ 0.1 – 0.2 H , rise with higher speed, v ≈ 1 – 2 km s − 1 , and dissipate right after rising to the corona without proceeding to the stage of further acceleration. Thick MFTs with a 0 = 0.4 H float with highest speeds up to 3 km s − 1 and dissipate near the surface of the disk.

Upward motion of the MFT is caused by the buoyancy force, which depends on the difference between internal and external densities, Δ ρ = ρ e − ρ . Figure 2(b) shows that Δ ρ > 0 and therefore the buoyant force is positive in all considered cases. The MFT expands and its density decreases during its motion in order to sustain the pressure balance. Near the surface of the disk and in the corona, Δ ρ approaches zero and therefore the MFT’s speed decreases. Abrupt deceleration of the MFT after passing the surface of the disk is caused by the abrupt disk’s density drop in this region of transition from the disk to corona.

The MFT stays in thermal equilibrium, T ≈ T e , during upward motion inside the disk, as shown in Figure 2(d). This is due to fast radiative heat exchange with the external gas. Departure form thermal equilibrium is observed only for the MFTs with a 0 = 0.1 − 0.2 H after their rising from the disk to the transition region, where T e grows up to the corona’s temperature of 475 K.

In Figure 3, we plot the dependence of the MFT’s speed on the z -coordinate at r = 0.5 au for z 0 = 0.5 H , a 0 = 0.1 H , and various initial plasma beta β 0 . Figure 3 shows that the MFTs with stronger magnetic field accelerate to greater speed. Maximum speed of 7–8 km s − 1 is achieved by the MFT with β 0 = 0.01 . The increase of the MFTs speed with β 0 is explained by the fact that the more initial magnetic field strength of the MFT, the more initial Δ ρ , and correspondingly the buoyant force is stronger. Closer to the star, r < 0.5 au , the maximum speed is of 10–12 km s − 1 , according to our simulations.

$Figure 3 Dependence of the MFT’s speed on the z z -coordinate for various initial plasma beta β 0 {\beta }_{0} in runs without external magnetic field. Vertical line delineates the surface of the disk. Initial parameters: r = 0.5 au r=0.5\hspace{0.33em}{\rm{au}} , z 0 = 0.5 H {z}_{0}=0.5\hspace{0.33em}H , and a 0 = 0.1 H {a}_{0}=0.1\hspace{0.33em}H .$

Figure 3

Dependence of the MFT’s speed on the z -coordinate for various initial plasma beta β 0 in runs without external magnetic field. Vertical line delineates the surface of the disk. Initial parameters: r = 0.5 au , z 0 = 0.5 H , and a 0 = 0.1 H .

3.3 Magnetic oscillations

In this section, we investigate how does the magnetic pressure outside the MFT influences its dynamics. In this case, external pressure P e in (6) is a sum of gas pressure and magnetic pressure B e 2 / 8 π . We assume that B e is constant with z and has magnitude of B z .

In Figure 4, we present simulation results for the MFT at r = 0.5 au with fiducial z 0 = 0.5 H , a 0 = 0.1 H , and various initial plasma beta, β 0 . The dependences of MFT’s speed, density, radius, and temperature on the z -coordinate are depicted. Magnetic field strength is of 8 G at considered r . Figure 4 shows that the dynamics of the MFT differs from the case with zero external magnetic field. The MFT floats with acceleration up to some height z o near the surface of the disk, and then its motion becomes oscillatory: the MFT moves vertically up and down around the point z o . According to Figure 4(a), z o ≈ 2.2 , 2.5, and 2.6 H for β 0 = 1 , 0.1, and 0.01, respectively. Hence, the more the β 0 , the higher the point z o , around which the MFT oscillates. Magnitude of the MFT’s speed decreases during oscillations as the MFT loses its kinetic energy due to the friction with the external gas.

$Figure 4 Dynamics of MFTs with various initial plasma beta β 0 {\beta }_{0} in the presence of external magnetic field. Dependences of the MFT’s speed (panel a), density (b), cross-section radius (c), and temperature (d) on the z z -coordinate are plotted. Vertical lines show the surface of the disk. Grey dashed lines in panels (b) and (d) delineate corresponding profiles of the disk density and temperature. Initial parameters: r = 0.5 au r=0.5\hspace{0.33em}{\rm{au}} , z 0 = 0.5 H {z}_{0}=0.5\hspace{0.33em}H , and a 0 = 0.1 H {a}_{0}=0.1\hspace{0.33em}H .$

Figure 4

Dynamics of MFTs with various initial plasma beta β 0 in the presence of external magnetic field. Dependences of the MFT’s speed (panel a), density (b), cross-section radius (c), and temperature (d) on the z -coordinate are plotted. Vertical lines show the surface of the disk. Grey dashed lines in panels (b) and (d) delineate corresponding profiles of the disk density and temperature. Initial parameters: r = 0.5 au , z 0 = 0.5 H , and a 0 = 0.1 H .

When the MFT starts to oscillate, its expansion stops at some characteristic cross-section radius a o . In the considered case, this radius is of 0.3 H , according to Figure 4(c). The radius of the MFT periodically increases and decreases with respect to a o during the oscillations, i.e., the MFT pulsates. The magnitude of the radius variations decreases, i.e., the pulsations decay with time.

Figure 4(b) shows that the point z o is a point of zero buoyancy, such that Δ ρ = ρ e − ρ > 0 at z < z o and Δ ρ < 0 at z > z o . This effect is caused by the contribution of the magnetic pressure outside the MFT to the overall pressure balance (6). The external magnetic field B e is constant with z , while the density of the disk ρ e exponentially decreases. As a consequence, the magnetic pressure B e 2 / 8 π contribution to P e also increases in comparison to the gas pressure. At z ≥ z o , the external magnetic field becomes stronger than the magnetic field of the MFT, and consequently the MFT becomes heavier than the external gas. This result is similar to that found by Dudorov et al. (2019) for the MFT in the accretion disks of TTS.

The beginning of the magnetic oscillations is characterized by violation of the thermal balance, T ≠ T e , as shown in Figure 4(d). This means that the rate of radiative heat exchange is smaller than the rate of MFT’s cooling due to adiabatic expansion. During the oscillations, the MFT’s pulsations decay and radiative heat exchange ultimately equalizes the temperature T and T e .

In Figure 5, we plot the corresponding dependences of MFT’s temperature on time. Figure 5 clearly demonstrates the periodic changes in MFT’s temperature during the magnetic oscillations. The period of oscillations increases with β 0 and lies in range from 0.5 month for β 0 = 0.01 to 2 months for β 0 = 1 .

$Figure 5 Dependence of the MFT’s temperature on time for various initial plasma beta β 0 {\beta }_{0} in presence of external magnetic field. Horizontal dashed line the delineates the temperature of the disk’s corona. Initial parameters: r = 0.5 au r=0.5\hspace{0.33em}{\rm{au}} , z 0 = 0.5 H {z}_{0}=0.5\hspace{0.33em}H , and a 0 = 0.1 H {a}_{0}=0.1\hspace{0.33em}H .$

Figure 5

Dependence of the MFT’s temperature on time for various initial plasma beta β 0 in presence of external magnetic field. Horizontal dashed line the delineates the temperature of the disk’s corona. Initial parameters: r = 0.5 au , z 0 = 0.5 H , and a 0 = 0.1 H .

Picture of the MFT’s thermal evolution in the case β 0 = 1 can be described as simple decaying oscillations. In this case, the oscillations take place under the surface of the disk (Figure 4(a)), and the MFT’s temperature follows the polytropic disk’s temperature profile during its periodic upward and downward motion. The MFTs with β 0 ≤ 0.1 exhibit more complex behavior characterized by non-harmonic oscillations of the temperature. In the case β 0 = 0.01 , the maximum of each temperature pulsation is characterized by a constant T = 475 K during time interval of 0.1–0.5 months. Such a behavior is explained by the fact that the MFTs with β 0 ≤ 0.1 oscillate near the surface of the disk. The vertical profile of disk’s temperature is non-monotonic in this region with minimum at z s , according to Figure 4(d). Therefore, the maximum T in the oscillating MFT corresponds to the corona’s temperature of 475 K, while the minimum T is achieved at some point below the surface of the disk.

In order to investigate characteristic time scales of this process, we plot the dependence of the z -coordinate, temperature, and magnetic field strength of the MFT on time in Figure 6. The results for the MFT with fiducial parameters z 0 = 0.5 H , a 0 = 0.1 H , and β 0 = 1 at various radial distances from the star are shown. Considered radial distances are marked with empty circles in T ( r ) , Σ ( r ) , x ( r ) , and B ( r ) profiles in Figure 1.

$Figure 6 Dynamics of the MFTs at different r r in the presence of external magnetic field. Top row: the dependence of the z z -coordinate of the MFT on time during its motion inside the disk at various radial distances r = 0.012 r=0.012 , 0.15, and 1 au (panels a, b, and c, respectively). Horizontal lines show the surface of the disk. Bottom row (panels d, e, and f, respectively): corresponding dependences of temperature (left y y -axis, black lines) and magnetic field strength (right y y -axis, blue lines) on time. Horizontal dashed blue lines correspond to the external magnetic field B e {B}_{{\rm{e}}} . Initial radius and plasma beta of the MFT are a 0 = 0.1 H {a}_{0}=0.1\hspace{0.33em}H and β 0 = 1 {\beta }_{0}=1 , respectively.$

Figure 6

Dynamics of the MFTs at different r in the presence of external magnetic field. Top row: the dependence of the z -coordinate of the MFT on time during its motion inside the disk at various radial distances r = 0.012 , 0.15, and 1 au (panels a, b, and c, respectively). Horizontal lines show the surface of the disk. Bottom row (panels d, e, and f, respectively): corresponding dependences of temperature (left y -axis, black lines) and magnetic field strength (right y -axis, blue lines) on time. Horizontal dashed blue lines correspond to the external magnetic field B e . Initial radius and plasma beta of the MFT are a 0 = 0.1 H and β 0 = 1 , respectively.

Figure 6(a, b, and c) show that the magnetic oscillations take place beneath the surface of the disk, at z ∼ 2 – 2.5 H typically. The oscillations are found at every radial distance in the considered range, r = 0.012 – 1 au . The period of oscillations P o increases with r , such that P o ≈ 0.2 d ≈ 5 h at r = 0.012 au, 2 months at r = 0.5 au, and 5 months at r = 1 au. The amplitude of upward and downward motion decreases with time.

According to Figure 6(d, e, and f), the magnetic oscillations are accompanied by the corresponding periodic changes in the MFT’s temperature. The MFT heats up during the period of downward motion and cools down during its upward motion. These changes reflect the z -distribution of the disk’s temperature T e , since the radiative heat exchange tends to keep the MFT in thermal balance with the external gas. The magnitude of the temperature fluctuations decreases with r . In maximum, it ranges from Δ T ∼ 3,000 K at r = 0.012 au to 300 K at r = 1 au.

Dependences B ( t ) depicted in Figure 6(d, e, and f) show that the MFT’s magnetic field strength decreases during its upward motion up to a point of the zero buoyancy. This decrease reflects the expansion of the MFT, B ∝ a − 2 according to the magnetic flux conservations. During the following magnetic oscillations, the magnetic field strength periodically increases and decreases with respect to the value B e . This picture confirms above discussions that the magnetic oscillations arise due to the effect of external magnetic pressure near the point of zero buoyancy, which is characterized by the equality B ≈ B e .

4 Conclusions and discussion

We numerically modeled the dynamics of MFTs in the accretion disk of typical HAeBeS. The simulations were carried out in frame of the slender flux tube approximations using the model developed by Dudorov et al. (2019). This model allows us to investigate the motion of the MFT in the direction perpendicular to the disk’s plane taking into account the buoyant and drag forces, radiative heat exchange of the MFT with external gas, and magnetic pressure of the disk.

The structure and characteristics of the accretion disk were calculated using our MHD model of the accretion disks (see Khaibrakhmanov et al. 2017), which is based on the model of Shakura and Sunyaev (1973). The vertical structure of the disk at each radial distance r is calculated from the solution of the hydrostatic equilibrium equation for the case of polytropic gas. It is considered that the turbulent friction is the main heating mechanism inside the disk. There is optically thin corona above the optically thick disk. The temperature of the corona is determined by the heating of the gas with incident stellar radiation. We adopted that the fraction of the radiation flux intercepted by the disk’s surface is constant, f = 0.05 , at every r . Transition from the disk to its corona is treated as the hydrostatic region with exponential growth of gas temperature over the local scale height of the disk.

We adopted the parameters of the star and its accretion disk corresponding to the star MWC 480. This is a typical “isolated” HAeBeS, which was investigated in detail in different spectral ranges (see Sitko et al. 2008, Mendigutía et al. 2013, Tambovtseva et al. 2016, Fernandez et al. 2018).

Our simulations have shown that the accretion disk of the HAeBeS is in general larger, denser, and hotter than the accretion disk of typical TTS. This is because the disk in the former case is characterized by larger accretion rate. As a consequence, the magnetic field in the disk of HAeBeS is stronger than in the disk of TTS. The innermost region of the disk, where temperature is high enough for thermal ionization of alkali metals and hydrogen and where the magnetic field is frozen into gas, is more extended in the case of HAeBeS. This region ranges from 0.012 au up to r = 1 au in the radial direction, under adopted parameters.

We modeled the dynamics of the MFT of various initial cross-section radii, a 0 , and plasma beta, β 0 , at several radial distances r in the range 0.012–1 au. The simulations have shown that the dynamics of the MFT in the accretion disk of the HAeBeS is in general qualitatively similar to the case of typical TTS. In the absence of the external magnetic field, MFTs rise from the disk with typical speeds up to 10–12 km s − 1 and form outflowing magnetized corona of the disk. Radiative heat exchange rapidly equalizes the temperatures inside and outside the MFT, T ≈ T e , under all considered parameters. We did not find thermal oscillations of the MFT, caused by adiabaticity, unlike the case of the TTS, for which the thermal oscillations of the MFT with a 0 ∼ 0.1 and β 0 = 1 at r < 0.2 au were found by Dudorov et al. (2019). Like in the case of TTS, MFTs transport excess of the disk’s magnetic flux into its corona.

The pressure of the magnetic field outside the MFT halts upward motion of the MFT near the point, where internal and external magnetic fields are nearly equal. This point of zero buoyancy, z o , typically lies near the surface of the disk, z s ∼ 2.5 – 3 H , where H is the local isothermal scale height. The more the initial magnetic field strength of the MFT, the higher the point z o lies. After the MFT rises to this point, its motion becomes oscillatory. The MFT moves up and down around the point z o and pulsates. The magnitude of these magnetic oscillations decreases with time because of the loss of the MFT’s kinetic energy due to friction with external gas. The period of the oscillations increases with radial distance and ranges from few hours at the inner boundary of the disk, r = 0.012 au, up to several months at r = 1 au in the case of typical a 0 = 0.1 H and β 0 = 1 . The oscillation period increases with β 0 at a given r .

Correspondingly, the temperature of the MFT experiences decaying oscillations around the value of local external temperature at the point z o . During the first few periods of the oscillations, temperatures inside and outside the MFT are not equal to each other, i.e., radiative heat exchange is not efficient, and the MFT is practically adiabatic. But ultimately the heat exchange equalizes temperature of the MFT and external gas. The maximum magnitude of the temperature variations ranges from several thousand Kelvin at the inner edge of the disk to several hundred Kelvin at r = 1 au . Temperature variations during each period of the oscillations may be non-harmonic and asymmetric, since the oscillations take place near the surface of the disk, where the dependence of the disk’s temperature on the z -coordinate is complex and non-monotonic.

Following original idea of Khaibrakhmanov et al. (2018), we propose that the oscillations of MFTs can be a source of the emission variability as well as variable circumstellar extinction observed in young stars with accretion disks. Such a variability is a widespread feature of the accretion disks of TTS and HAeBeS (see Kóspál et al. 2021, Flaherty et al. 2016), which has also been found for the MWC 480 star considered as a reference in our modeling. Generally speaking, periodically rising and oscillating MFTs could contribute to the variability of the emission in different spectral ranges emanating from the innermost region of the disk, where the magnetic field is frozen in into gas: r = 0.012 – 1 au in the case of considered HAeBeS. The MFTs formed beyond the dust sublimation radius, r = 0.5 au , could contain dust grains. In this case, temperature fluctuations of the oscillating MFT may cause the IR-variability of the disk. This assumption is supported by the observations of MWC 480 demonstrating the variations in the IR-flux over 1 – 13 μ m wavelength range (Sitko et al. 2008, Fernandez et al. 2018). This radiation emanates from the dust sublimation zone and points to the changes of the disk’s structure in this region. It should be noted that inhomogeneities in the disk centrifugal winds containing both gas and dust can cause the variability of young stars’ emission, in particular the variations in circumstellar extinction observed in young stars (Tambovtseva and Grinin 2008). Application of both models to specific systems with well-established variability is needed in order to determine the relative role of various variability mechanisms.

In general, our results have shown that the accretion disks of HAeBeS have more extended region of the efficient generation of the magnetic field than in the case of TTS. The temperature of their corona is higher due to more intense stellar radiation. As a consequence, temperature variations in the oscillating MFTs have larger magnitude. Therefore, the IR-variability caused by oscillating MFTs would be more intense in the case of accretion disks of HAeBeS as compared to TTS.

In order to investigate the connection between magnetic oscillations of MFTs and IR-variability of TTS and HAeBeS, we plan to calculate spectral energy distributions of the accretion disks taking into account variations of their structure due to the effect of rising MFTs. Interesting task is to model the synthetic light-curves of the accretion disks taking into account contribution of periodically rising MFTs into the IR flux of the disk.

Acknowledgements

The authors thank anonymous referee for useful comments.

Funding information: This work was supported by the Russian Science Foundation (project 19-72-10012)
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-11-01

Revised: 2022-02-02

Accepted: 2022-02-09

Published Online: 2022-03-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/astro-2022-0017

Keywords for this article

accretion discs; MHD; ISM: magnetic fields; stars: variables: T Tauri; Herbig Ae/Be

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