Rotational Disruption of Dust and Ice by Radiative Torques in Protoplanetary Disks and the Implications for Observations

For our calculations, we adopt a two-dimensional (2D) flared, radiative, and hydrostatic equilibrium disk model (Chiang & Goldreich 1997) as illustrated in Figure 1. The disk surface is defined by a slant path of unity optical depth of τ_V ∼ 1. This layer is directly heated by stellar radiation, and is considered a photodissociated region. Dust and gas below this layer are heated by attenuated stellar radiation and define a warm layer ($T\sim 30\mbox{--}100\,{\rm{K}})$. The final region is the disk interior, which is completely shielded from stellar radiation and only heated by infrared emission from hot dust in the surface layer.

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The gas number density at disk radius R and height z for the hydrostatic disk model, assuming a Gaussian vertical profile (Lynden-Bell & Pringle 1974), is given by

$$\begin{eqnarray}&&{n}_{{\rm{H}}}(R,z)\approx \displaystyle \frac{1}{2{m}_{{\rm{H}}}}\displaystyle \frac{{\rm{\Sigma }}(R)}{{H}_{p}\sqrt{2\pi }}\exp \left(-\displaystyle \frac{{z}^{2}}{2{H}_{p}^{2}}\right),\end{eqnarray} \tag{ 1 }$$

where Σ(R) is the total surface mass density at radius R given by

$$\begin{eqnarray}&&{\rm{\Sigma }}(R)={{\rm{\Sigma }}}_{1}{\left(\displaystyle \frac{R}{1\mathrm{au}}\right)}^{-\alpha },\end{eqnarray} \tag{ 2 }$$

where α is the model constant, and Σ₁ is the surface mass density at $R=1\,\mathrm{au}$.

The pressure height scale H_p is described by

$$\begin{eqnarray}&&\displaystyle \frac{{H}_{p}}{R}=\displaystyle \frac{{H}_{0}}{{R}_{0}}{\left(\displaystyle \frac{R}{{R}_{0}}\right)}^{1/7},\end{eqnarray} \tag{ 3 }$$

where H₀ is the aspect ratio at the reference radius R₀.

For R₀ = 100 au, H₀/R₀ is taken to be 0.1 as a fiducial model, which corresponds to H_p/R = 0.1 × 3^1/7 at R_out = 300 au. The chosen aspect ratio is much lower than predicted by Chiang & Goldreich (1997), but is comparable to observations (Avenhaus et al. 2018). Here, we assume α = 1 and Σ₁ is varied to cover a wide range of disk mass. Other physical parameters are listed in Table 1, including the stellar temperature T_⋆, stellar mass M_⋆, stellar radius R_⋆, and the inner and outer disk radii R_in and R_out.

Table 1.  Physical Parameters of a PPD

Objects	T⋆	M⋆	R⋆	Rin	Rout
	$({\rm{K}})$	(M⊙)	(R⊙)	$(\mathrm{au})$	$(\mathrm{au})$
Herbig Ae/Be	10000	2	2	1	300
T-Tauri	4000	0.5	2	0.1	300

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With the disk model defined, we compute the grain temperature throughout the disk by performing a radiative transfer calculation using the publicly available 3D Monte Carlo radiative transfer code (radmc-3d; Dullemond et al. 2012).⁶ The grid resolutions are N_r = 128 for R_in < R < R_out and N_θ = 128 for 0 < θ < 2π. The number of photons is N_phot ∼ 10⁹. The dust opacity is calculated assuming a power-law size distribution, n(a) ∝ a^−q (a_min < a < a_max) with the absorption cross section for spherical grains computed using the Mie theory coded from Bohren & Huffman (1983), assuming the optical constant of amorphous silicate (Mg_0.7Fe_0.3SiO₃).⁷ We adopt lower and upper cutoffs for the size distribution of large grains a_min = 0.1 μm and a_max = 10 μm, and the standard MRN distribution q = 3.5 (Mathis et al. 1983). The dust has a total mass ranging from 2 × 10⁻⁶ to 2 × 10⁻³M_⊙ for ${{\rm{\Sigma }}}_{1}=1\mbox{--}1000\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and the same vertical structure as the gas.

Figures 2 and 3 show the gas density and grain temperature from radmc-3d for the disks around T-Tauri and Herbig Ae/Be stars, respectively, with ${{\rm{\Sigma }}}_{1}=1000\,{\rm{g}}\,{\mathrm{cm}}^{-2}$.

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The mean wavelength of the radiation field of a young star can be calculated by

$$\begin{eqnarray}&&\bar{\lambda }=\displaystyle \frac{\int \lambda {u}_{\lambda }({T}_{\star })d\lambda }{\int {u}_{\lambda }({T}_{\star })d\lambda },\end{eqnarray} \tag{ 4 }$$

where the radiation intensity u_λ (T_⋆) is described by the Planck function. This yields $\overline{\lambda }=1.33\,\mu {\rm{m}}$ for the T-Tauri disks and $\overline{\lambda }=0.53\,\mu {\rm{m}}$ for the Herbig Ae/Be disks. The radiation energy density is ${u}_{\mathrm{rad}}=\int {u}_{\lambda }d\lambda$.

The strength of the radiation field in the disk is characterized by a dimensionless parameter U = u_rad/u_ISRF where ${u}_{\mathrm{ISRF}}\,=8.64\times {10}^{-13}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ is the energy density of the average interstellar radiation field (ISRF) in the solar neighborhood from Mathis et al. (1983). The local value of U in the disk can be approximately calculated using the grain temperature obtained from radmc-3d as follows:

$$\begin{eqnarray}&&U\simeq {\left(\displaystyle \frac{a}{0.1\mu {\rm{m}}}\right)}^{6/15}{\left(\displaystyle \frac{{T}_{d}}{16.4{\rm{K}}}\right)}^{6},\end{eqnarray} \tag{ 5 }$$

for silicate grains (see Draine 2011).

When exposed to an anisotropic radiation field, a dust grain of irregular shape experiences RATs (Dolginov & Mitrofanov 1976) that can spin it up to suprathermal rotation (Draine & Weingartner 1996; Hoang & Lazarian 2009). Numerical calculations have revealed that RATs are weakly dependent on the dust composition (Lazarian & Hoang 2007; Herranen et al. 2019). Therefore, the following discussion is applied to both carbonaceous, silicate grains as well as carbon-silicate mixtures.

Let a be the effective grain size, which is defined as the radius of the equivalent sphere of the same volume. Following Hoang (2019), the maximum rotation rate of dust grains spun-up by a radiation field of anisotropy γ, mean wavelength $\bar{\lambda }$, and radiation strength U is given by

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{\mathrm{RAT}} & \simeq & 9.6\times {10}^{8}\gamma {a}_{-5}^{0.7}{\bar{\lambda }}_{0.5}^{-1.7}\\ & & \times \ \left(\displaystyle \frac{U}{{n}_{1}{T}_{2}^{1/2}}\right)\left(\displaystyle \frac{1}{1+{F}_{\mathrm{IR}}}\right)\,\mathrm{rad}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 6 }$$

for grains with $a\lesssim \bar{\lambda }/1.8$, and

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{\mathrm{RAT}} & \simeq & 1.78\times {10}^{10}\gamma {a}_{-5}^{-2}{\bar{\lambda }}_{0.5}\\ & & \times \ \left(\displaystyle \frac{U}{{n}_{1}{T}_{2}^{1/2}}\right)\left(\displaystyle \frac{1}{1+{F}_{\mathrm{IR}}}\right)\,\mathrm{rad}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 7 }$$

for grains with $a\gt \overline{\lambda }/1.8$. Here, ${n}_{1}={n}_{{\rm{H}}}/(10\,{\mathrm{cm}}^{-3})$, ${T}_{2}\,={T}_{\mathrm{gas}}/(100\,{\rm{K}})$, ${\bar{\lambda }}_{0.5}=\bar{\lambda }/(0.5\,\mu {\rm{m}})$, ${a}_{-5}=a/({10}^{-5}\,\mathrm{cm})$, and F_IR is the dimensionless parameter describing the grain rotational damping by infrared emission that depends on n_H, T_gas, and U (Draine & Lazarian 1998; Hoang et al. 2010). The rotation rate depends on the parameter $U/{n}_{{\rm{H}}}{T}_{\mathrm{gas}}^{1/2}$ and the damping by far-infrared emission F_IR.

For convenience, let ${a}_{\mathrm{trans}}=\bar{\lambda }/1.8$, which denotes the grain size at which the RAT efficiency changes between the power-law and flat stages (see, e.g., Lazarian & Hoang 2007; Hoang et al. 2019) and ω_RAT changes from Equation (6) to (7).

The centrifugal force applied to a spinning grain induces centrifugal stress S = ρω²a²/4, with ρ being the mass density of the grain; this stress tends to tear the grain apart. When the centrifugal stress exceeds the maximum tensile strength of the grain material, the grain is instantaneously disrupted into small fragments. This mechanism of dust destruction, developed by Hoang et al. (2019), is termed RATD.

The critical rotation rate required to disrupt the composite grain is described by

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{\mathrm{disr}} & = & \displaystyle \frac{2}{a}{\left(\displaystyle \frac{{S}_{\max }}{\rho }\right)}^{1/2}\\ & \simeq & \ \displaystyle \frac{3.6\times {10}^{8}}{{a}_{-5}}{S}_{\max ,7}^{1/2}{\hat{\rho }}^{-1/2}\,\,\mathrm{rad}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 8 }$$

where ${S}_{\max ,7}={S}_{\max }/({10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$), with S_max being the maximum tensile strength of the grain and $\hat{\rho }=\rho /(3\,{\rm{g}}\,{\mathrm{cm}}^{-3})$ (Hoang et al. 2019).

The value of S_max depends on the composition, internal structures, and grain size. In PPDs, dust particles can have a large range of sizes, from nanoparticles to mm/cm sized pebbles, owing to the effect of grain coagulation and growth. To model RATD in the surface and intermediate layers of PPDs, we consider the size dependence of the tensile strength. Grains of size a > 0.1 μm can be considered as porous/composite. For these grains, we adopt a composite grain model as proposed by Mathis & Whiffen (1989), in which individual particles of silicate or carbonaceous materials are assumed to be compact and spherical of radius a_p.

Following Greenberg et al. (1995), the maximum tensile strength of the grain is given by

$$\begin{eqnarray}&&{S}_{\max }=3\beta (1-P)\displaystyle \frac{\bar{E}}{2{{ha}}_{p}^{2}},\end{eqnarray} \tag{ 9 }$$

where $\bar{E}=\alpha {10}^{-3}\,\mathrm{eV}\,$ is the mean intermolecular interaction energy at the contact surface between two particles, h is the mean intermolecular distance, P is the porosity of the grain, and β is the mean number of contact points per particle between 1 and 10. For our estimates, we fix the porosity P = 0.2, as previously assumed for Planck data modeling (Guillet et al. 2018), and adopt the typical values α = 1, β = 5, h = 0.3 nm. Equation (9) yields ${S}_{\max }=3.2\times {10}^{5}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ for a_p = 10 nm.

Grains of a ≤ 0.05 μm are likely compact and have higher S_max than that of composite grains. We take the typical value ${S}_{\max }={10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ for these grains. For grain sizes in between ($0.1\,\mu {\rm{m}}\geqslant a\gt 0.05\,\mu {\rm{m}}$), we assume an intermediate value of ${S}_{\max }={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$.

Grain disruption sizes can be computed by solving the equation ω_RAT = ω_disr. From Equation (8), one can see that ω_disr decreases as ∝a⁻¹, whereas Equations (6) and (7) imply that ω_RAT first increases with a until a = a_trans and then decreases more rapidly than ω_disr as a goes beyond a_trans. Hence, one can expect there are two intersections of ω_RAT and ω_disr: one for a < a_trans and one for a > a_trans. Upon reaching the critical size determined by the first intersection of ω_RAT and ω_disr, grains are disrupted. This value can be obtained from Equations (6) and (8), as given by

$$\begin{eqnarray}\begin{array}{rcl}{a}_{\mathrm{disr}} & \simeq & 0.06{\gamma }^{-1/1.7}{\bar{\lambda }}_{0.5}{\left({S}_{\max ,7}/\hat{\rho }\right)}^{1/3.4}\\ & & \times \ {\left(1+{F}_{\mathrm{IR}}\right)}^{1/1.7}{\left(\displaystyle \frac{{n}_{1}{T}_{2}^{1/2}}{U}\right)}^{1/1.7}\,\mu {\rm{m}},\end{array}\end{eqnarray} \tag{ 10 }$$

for a ≲ a_trans. The maximum size of grains that can still be disrupted is determined by the second intersection and can be estimated from Equations (7) and (8), as given by

$$\begin{eqnarray}\begin{array}{rcl}{a}_{\mathrm{disr},\max } & \simeq & 4.9\gamma {\bar{\lambda }}_{0.5}{\hat{\rho }}^{1/2}{S}_{\max ,7}^{-1/2}\\ & & \times \ \left(\displaystyle \frac{1}{1+{F}_{\mathrm{IR}}}\right)\left(\displaystyle \frac{U}{{n}_{1}{{T}_{2}}^{1/2}}\right)\,\mu {\rm{m}},\end{array}\end{eqnarray} \tag{ 11 }$$

which depends on the local gas density and temperature due to gas damping. Under the effect of rotational disruption, all grains within the size range a_disr ≤ a ≤ a_disr,max would be disrupted.

In the absence of rotational damping, the time required to spin-up the grains of size a_disr to ω_disr defines the disruption time:

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{disr}} & = & \displaystyle \frac{I{\omega }_{\mathrm{disr}}}{{dJ}/{dt}}=\displaystyle \frac{I{\omega }_{\mathrm{disr}}}{{{\rm{\Gamma }}}_{\mathrm{RAT}}}\\ & \simeq & \ {\left(\gamma {U}_{5}\right)}^{-1}{\bar{\lambda }}_{0.5}^{1.7}{\hat{\rho }}^{1/2}{S}_{\max ,7}^{1/2}{\left(\displaystyle \frac{{a}_{\mathrm{disr}}}{0.1\mu {\rm{m}}}\right)}^{-0.7}\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 12 }$$

where U₅ = U/10⁵.

When rotational damping is present, solving the equation ω(t) = ω_disr yields the disruption timescale

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{disr}} & = & -{\tau }_{\mathrm{damp}}\mathrm{ln}\left(1-\displaystyle \frac{{\omega }_{\mathrm{disr}}}{{\omega }_{\mathrm{RAT}}}\right)\\ & = & \ -{\tau }_{\mathrm{damp}}\mathrm{ln}\left(1-\displaystyle \frac{{t}_{\mathrm{disr},0}}{{\tau }_{\mathrm{damp}}}\right),\end{array}\end{eqnarray} \tag{ 13 }$$

which is applicable for a_disr,max > a > a_disr. Note that, for a = [a_disr, a_disr,max], ${t}_{\mathrm{disr}}\to \infty$ because it takes t ≫ t_damp to reach ω = ω_RAT. In strong radiation fields, when t_disr,0 ≪ τ_damp, t_disr returns to t_disr,0.

The rotational desorption mechanism of icy grain mantles was first studied by Hoang & Tram (2020), who found that the centrifugal stress induced by grain suprathermal rotation can desorb the entire ice mantle into tiny icy fragments, provided that the grain core is compact (see Figure 4).

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The grain model considered in Hoang & Tram (2020) is the one made of an amorphous silicate/carbonaceous core covered by a thick double-layer ice mantle (Oberg et al. 2010), where complex organic molecules are formed as illustrated in Figure 4. It is worth noting that the formation of ice mantles takes place on the grain surface, due to accretion of gas molecules in cold and dense regions of hydrogen density ${n}_{{\rm{H}}}=n({\rm{H}})+2n({{\rm{H}}}_{2})\sim {10}^{3}\mbox{--}{10}^{5}\,{\mathrm{cm}}^{-3}$ or the visual extinction A_V > 3 (Whittet et al. 1983). Although icy grain mantles have nonspherical shapes, as demonstrated by dust polarization, let us assume that the grain shape can be described by an equivalent sphere of the same volume with effective radius a.

As discovered in Hoang & Tram (2020), the centrifugal stress applied to a spinning core-ice mantle grain tends to pull off the ice mantle from the grain core at sufficiently fast rotation. Upon reaching the critical disruption limit ω_disr, the centrifugal stress is equal to the tensile strength that the mantle exerts upon the grain core, which causes the ice mantle near the equator to detach. When the rotation rate increases beyond ω_disr such that the centrifugal stress exceeds the ice tensile strength that holds different parts of the mantle together, the mantle is disrupted into small fragments.

Let x₀ be the distance from the core–mantle interface to the spinning axis. From Equation (7) in Hoang & Tram (2020), one obtains the centrifugal stress on the ice mantle:

$$\begin{eqnarray}&&{S}_{x}\simeq 2.5\times {10}^{9}{\hat{\rho }}_{\mathrm{ice}}{\omega }_{10}^{2}{a}_{-5}^{2}\left[1-{\left(\displaystyle \frac{{x}_{0}}{a}\right)}^{2}\right]\,\mathrm{erg}\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 14 }$$

where ${\hat{\rho }}_{\mathrm{ice}}={\rho }_{\mathrm{ice}}/(1\,{\rm{g}}\,{\mathrm{cm}}^{-3})$ with ρ_ice being the mass density of the ice mantle, which is ${\rho }_{\mathrm{ice}}\sim 1\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ for pure ice (Hoang & Tram 2020), and ${\omega }_{10}=\omega /({10}^{10}\,\mathrm{rad}\,\,{{\rm{s}}}^{-1})$.

The critical rotational velocity for disruption is given by S_x = S_max:

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{\mathrm{disr}} & = & \displaystyle \frac{2}{a{\left(1-{x}_{0}^{2}/{a}^{2}\right)}^{1/2}}{\left(\displaystyle \frac{{S}_{\max }}{{\rho }_{\mathrm{ice}}}\right)}^{1/2}\\ & \simeq & \ \displaystyle \frac{6.3\times {10}^{8}}{{a}_{-5}{\left(1-{x}_{0}^{2}\right)}^{1/2}}{\hat{\rho }}_{\mathrm{ice}}^{-1/2}{S}_{\max ,7}^{1/2}\,\mathrm{rad}\,{{\rm{s}}}^{-1}.\end{array}\end{eqnarray} \tag{ 15 }$$

Similar to rotational disruption, one can obtain the desorption sizes of icy grains, as given by

$$\begin{eqnarray}\begin{array}{rcl}{a}_{\mathrm{desp}} & \simeq & 0.13{\gamma }^{-1/1.7}{\bar{\lambda }}_{0.5}{\left({S}_{\max ,7}/{\hat{\rho }}_{\mathrm{ice}}\right)}^{1/3.4}\\ & & \times \ {\left(1+{F}_{\mathrm{IR}}\right)}^{1/1.7}{\left(\displaystyle \frac{{n}_{1}{T}_{2}^{1/2}}{U}\right)}^{1/1.7}\,\mu {\rm{m}},\end{array}\end{eqnarray} \tag{ 16 }$$

for $a\leqslant \overline{\lambda }/1.8$, and the maximum size of icy grains that can still be disrupted by centrifugal stress caused by RATs is described by

$$\begin{eqnarray}\begin{array}{rcl}{a}_{\mathrm{desp},\max } & \simeq & 2.9\gamma {\bar{\lambda }}_{0.5}{\hat{\rho }}_{\mathrm{ice}}^{1/2}{S}_{\max ,7}^{-1/2}\\ & & \times \ \left(\displaystyle \frac{1}{1+{F}_{\mathrm{IR}}}\right)\left(\displaystyle \frac{U}{{n}_{1}{T}_{2}^{1/2}}\right)\,\mu {\rm{m}}.\end{array}\end{eqnarray} \tag{ 17 }$$

In the absence of rotational damping, the characteristic timescale for rotational desorption can be estimated as:

$$\begin{eqnarray}&&{t}_{\mathrm{desp}}\simeq 0.6{\left(\gamma {U}_{5}\right)}^{-1}{\bar{\lambda }}_{0.5}^{1.7}{\hat{\rho }}_{\mathrm{ice}}^{1/2}{S}_{\max ,7}^{1/2}{\left(\displaystyle \frac{{a}_{\mathrm{desp}}}{0.1\mu {\rm{m}}}\right)}^{-0.7}\,\mathrm{yr}.\end{eqnarray} \tag{ 18 }$$

In addition to rotational desorption, water and complex molecules can also be desorbed from icy grain mantles by the so-called rotational-thermal or ro-thermal desorption mechanism. Note that, in star-forming regions, molecules are assumed to be physically adsorbed to the ice mantle via the van der Waals force, also referred to as physisorption. In this mechanism, instead of evaporating from the disrupted icy fragments of the original mantle as studied in the previous subsection, molecules sublimate directly from the intact icy grain mantle with the support of centrifugal force. A detailed description and formulation of the ro-thermal desorption mechanism is presented in Hoang & Tung (2019). Here, we briefly describe the mechanism with regard to its application to the disks.

Let τ_sub,0 be the desorption rate of absorbed molecules with binding energy E_b from a grain at rest (ω = 0), which is heated to temperatures T_d. Following Watson & Salpeter (1972), one has

$$\begin{eqnarray}&&{\tau }_{\mathrm{sub},0}^{-1}=\nu \exp \left(-\displaystyle \frac{{E}_{b}}{{{kT}}_{d}}\right),\end{eqnarray} \tag{ 19 }$$

where ν is the characteristic vibration frequency of adsorbed molecules in the perpendicular direction to the lattice surface and given by

$$\begin{eqnarray}&&\nu ={\left(\displaystyle \frac{2{N}_{s}{E}_{b}}{{\pi }^{2}m}\right)}^{1/2},\end{eqnarray} \tag{ 20 }$$

with N_s being the surface density of binding sites (Tielens & Allamandola 1987). Typically, N_s ∼ 2 × 10¹⁵ site $\,{\mathrm{cm}}^{-2}$.

Following Hoang & Tung (2019), in the presence of grain suprathermal rotation, the centrifugal force acting on the absorbed molecule of mass m at distance $r\sin \theta$ from the spinning axis is

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{\mathrm{cen}}=m{{\boldsymbol{a}}}_{\mathrm{cen}}=m{\omega }^{2}(x\hat{{\boldsymbol{x}}}+y\hat{{\boldsymbol{y}}}),\end{eqnarray} \tag{ 21 }$$

where the unit vectors ($\hat{{\boldsymbol{x}}},\hat{{\boldsymbol{y}}}$) describe the plane perpendicular to the spinning axis, and ${{\boldsymbol{a}}}_{\mathrm{cen}}=-{\rm{\nabla }}{\phi }_{\mathrm{cen}}$ is the centrifugal acceleration. Here, the centrifugal potential

$$\begin{eqnarray}&&{\phi }_{\mathrm{cen}}=\displaystyle \frac{1}{2}{\omega }^{2}{r}^{2}\sin {\theta }^{2}=\displaystyle \frac{{\omega }^{2}{a}^{2}}{3},\end{eqnarray} \tag{ 22 }$$

which gives the average centrifugal potential

$$\begin{eqnarray}&&\langle {\phi }_{\mathrm{cen}}\rangle =\displaystyle \frac{{\omega }^{2}{a}^{2}}{3},\end{eqnarray} \tag{ 23 }$$

where r ≡ a and $\langle {\sin }^{2}\theta \rangle =2/3$ is taken.

As a result, the effective binding energy of the absorbed molecule becomes

$$\begin{eqnarray}&&{E}_{b,\mathrm{rot}}={E}_{b}-m\langle {\phi }_{\mathrm{cen}}\rangle ,\end{eqnarray} \tag{ 24 }$$

The rate of ro-thermal desorption is then given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{sub},\mathrm{rot}}^{-1}=\nu \exp \left(-\displaystyle \frac{{E}_{b}-m\langle {\phi }_{\mathrm{cen}}\rangle }{{{kT}}_{d}}\right),\end{eqnarray} \tag{ 25 }$$

where the second exponential term describes the probability of desorption induced by centrifugal potential.

We first show the results for disruption of dust grains in the surface and intermediate layers, which is a top-down mechanism to reproduce PAHs/nanoparticles as observed via mid-IR emission.

4.1.1. Disruption Sizes

To compute the disruption sizes of grains by RATD for a disk, we numerically solve the equation ω_RAT = ω_disr. We consider the maximum grain size a_max = 10 μm. We adopt γ = 0.7 for the anisotropy degree of the radiation field, which is appropriate for the surface and intermediate layers that are illuminated mostly by the stellar and ISRFs.

Figure 5 shows the obtained grain disruption sizes for a disk around T-Tauri stars with different values of Σ₁ ranging from 1 to 1000. The gray shaded region indicates that no disruption occurs, due to the low radiation intensity and/or high gas density. Depending on the surface mass density of the disks, rotational disruption occurs in both the warm intermediate and surface layers with ${T}_{d}\gt 30\,{\rm{K}}$. Lower values of Σ₁ result in an optically thin disk in which the disruption of grains is more prominent. In the disk surface, where the gas density ${n}_{{\rm{H}}}\lesssim {10}^{9}\,{\mathrm{cm}}^{-3}$ and grain temperature ${T}_{d}\gt 150\,{\rm{K}}$ (see Figure 2) for which grains could be spun-up to extremely fast rotation, rotational disruption is very efficient. The grain disruption size a_disr decreases with the disk height z from ${a}_{\mathrm{trans}}\sim 0.74\,\mu {\rm{m}}$ to a minimum value ∼0.01 μm in the central region ($R\lesssim 0.2\,\mathrm{au}$), indicating that all grains with sizes larger than ∼10 nm would be disrupted in this region to form nanoparticles. In the warm layer, rotational disruption is still pronounced in the region where ${T}_{d}=50\mbox{--}150\,{\rm{K}}$ and ${n}_{{\rm{H}}}\lesssim {10}^{6}\,{\mathrm{cm}}^{-3}$, and near the upper boundary of the disk height, small grains of a ≳ 0.05 μm are already disrupted. For lower grain temperatures and higher gas density (i.e., ${T}_{d}=30\mbox{--}50\,{\rm{K}}$ and ${n}_{{\rm{H}}}\lesssim {10}^{2}\,{\mathrm{cm}}^{-3}$ for the case of ${{\rm{\Sigma }}}_{1}\,=1000\,{\rm{g}}\,{\mathrm{cm}}^{-2}$), rotational disruption becomes less efficient and only occurs at the surface layer with z/R = 0.4–0.5. Nevertheless, grains of a ≳ 0.1 μm are still destroyed under the effect of RATD at the height boundary. The maximum disruption size a_disr,max increases rapidly with z and exceeds the threshold a_max = 10 μm in the surface layer, which means only very large grains could survive. Overall, the closer to the central star, the smaller a_disr and the greater a_disr,max, which means that a broader range of grain sizes is disrupted by RATD.

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It is otherwise the same as Figure 5, but Figure 6 instead shows the results for the disk around Herbig Ae/Be stars. The active region of rotational disruption is larger due to higher temperatures in the surface layer. At $R\sim 30\,\mathrm{au}$, small grains of a ∼ 0.03 μm can still be disrupted compared to $R\sim 0.3\,\mathrm{au}$ in the case of T-Tauri disks. Beyond that radius, disruption size is ∼0.05 μm at the top of the scale height and remains relatively small up to the outer radius (0.01–0.05 μm).

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To better understand rotational disruption in the different layers of the disk, in Figure 7, we show the disruption sizes as a function of the disk height z for the different disk radii, with ${{\rm{\Sigma }}}_{1}=1\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. From Equations (10) and (11), one can see that a_disr decreases when n_H decreases and T_d and U ∼ ${T}_{d}^{6}$ increases, whereas the opposite tendency is expected for a_disr,max. As n_H exponentially decreases and T_d is significantly greater as one goes higher to the exposed surface layer (see Figures 2 and 3), a_disr declines while a_disr,max increases rapidly with z, making the range [a_disr, a_disr,max] larger. The discontinuity of the plots for a_disr at a = 0.1 μm and a = 0.05 μm originates from the three separate values assumed for S_max for three ranges of grain sizes. Generally speaking, a_disr mostly ceases at 0.05 μm, due to the large tensile strength ${S}_{\max }\sim {10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ of compact grain cores of $a\leqslant 0.05\,\mu {\rm{m}}$. Only near the central region (e.g., $R=1\,\mathrm{au}$) can disruption sizes be smaller than 0.05 μm. Note that the maximum grain size is set to be a_max = 10 μm, which accounts for the stop at 10 μm of a_disr,max. However, such large grains may not experience lower RATs, due to incoherent contributions of RATs from the different grain facets (see, e.g., Cho & Lazarian 2007). Since the calculations of RATs for very large grains of size $a\gt \bar{\lambda }/0.1$ are not yet available, due to computing limitations (see, e.g., Herranen et al. 2019), in the case of Herbig disks where $\bar{\lambda }/0.1=5.33\,\mu {\rm{m}}$, we plot a horizontal line where $a=\bar{\lambda }/0.1$, below which RATs have been previously calculated for irregular grains (Lazarian & Hoang 2007).

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4.1.2. Rotational Disruption Time versus Grain–Grain Collision Destruction Time

We now calculate the time it takes for rotational disruption to disrupt the grains t_disr in the disks from Equation (12), using the numerical results for a_disr and U from Section 4.1.1.

Other possible dust destruction mechanism is grain shattering caused by grain–grain collisions induced by radiation pressure (Hoang 2017). The mean time between two collisions defines the destruction time by grain–grain collisions

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{\mathrm{gg}} & = & \displaystyle \frac{1}{\pi {a}^{2}{n}_{\mathrm{gr}}{v}_{\mathrm{gg}}}=\displaystyle \frac{4\rho {{aM}}_{{\rm{g}}/{\rm{d}}}}{3{n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}_{\mathrm{gg}}}\\ & \simeq & \ 2.5\times {10}^{7}\hat{\rho }{a}_{-5}\left(\displaystyle \frac{30\,{\mathrm{cm}}^{-3}}{{n}_{{\rm{H}}}}\right)\left(\displaystyle \frac{1\,\mathrm{km}\,{{\rm{s}}}^{-1}}{{v}_{\mathrm{gg}}}\right)\mathrm{yr},\end{array}\end{eqnarray} \tag{ 26 }$$

where a single size a distribution with the gas-to-dust mass ratio M_g/d = 100 and the number density of dust grains n_gr is assumed, and v_gg is the relative velocity of grains. For the turbulence effect, one has v_gg = α^pc_s, with the turbulence parameters α = 0.01 and p = 1/2 (see, e.g., Dullemond & Dominik 2004) and c_s being the gas sound speed. Comparing Equation (26) with (12), one can see that grain–grain collisions require a much longer timescale to produce small grains compared to RATD.

In Figure 8, we compare the disruption time by RATD (left panel) with the shattering time by grain–grain collisions (right panel) for different grain sizes a for the T-Tauri disks, with ${{\rm{\Sigma }}}_{1}=1\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. In the surface layer, where the gas and dust density are relatively low, the shattering time is extremely high, as τ_gg is most governed by n_H. In contrast, the disruption time, which depends on disruption sizes, is much lower due to high radiation intensity from the central star that causes grains to disrupt more easily—and hence, a_disr becomes smaller.

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It is worth noting that, the higher the value of Σ₁, the more massive the disk would be. A massive PPD, particularly around a T-Tauri star of 0.5M_⊙, is likely to be gravitationally unstable. In such a case, internal heating from gravitational instability–driven spirals and/or turbulence could dominate stellar radiation, giving rise to a disk model different from the passive irradiated disk adopted in Section 2. Furthermore, the Gaussian vertical profile of the gas described by Equation (1) assumes a vertically isothermal temperature structure, whereas the temperature profile from radmc-3d has a vertically stratified structure due to stellar radiation. As a result, the disk is not in hydrostatic equilibrium. However, the disruption time given by Equation (12) is smaller than the dynamical timescale, which is defined as the shortest timescale on which the disk structure can vary:

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{D} & = & \displaystyle \frac{1}{{\rm{\Omega }}}={\left(\displaystyle \frac{{{GM}}_{\star }}{{R}^{3}}\right)}^{-1/2}\\ & \simeq & \ 159.2{\left(\displaystyle \frac{R}{100\mathrm{au}}\right)}^{3/2}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{-1/2}\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 27 }$$

where Ω is the Keplerian orbital period. The problem of disk instability is well-studied in the literature (see, e.g., Nakamoto & Nakagawa 1994; Umurhan & Shaviv 2005; Pfeil & Klahr 2019) and it is well-established that the thermal and viscous timescales are much larger than τ_D. Therefore, we expect that the disruption of grains would not be significantly affected in such events.

Now we move on to show our results for rotational disruption and desorption of ice mantles, which is relevant to the warm intermediate layer.

4.2.1. Desorption Sizes of Ice Mantles

We first consider a core-ice mantle grain model with a fixed core radius a_c = 0.05 μm and a mantle thickness Δa_m that can vary such that the grain size is a = a_c + Δa_m. Similar to disruption sizes, numerical calculations for desorption sizes are performed with conservative values for the tensile strength of the grain core ${S}_{\max ,\mathrm{core}}={10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ and of the ice mantle ${S}_{\max ,\mathrm{mantle}}={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$, which is comparable to the tensile strength of bulk ice (see, e.g., Litwin et al. 2012).

The results for the T-Tauri disk are plotted in Figure 9. The snowline (${T}_{d}=150\,{\rm{K}}$) obtained from radmc-3d is shown for comparison. The similar variation features as rotational disruption can be observed here. Rotational desorption of ice mantles also occurs in the warm intermediate and surface layers with ${T}_{d}\gt 30\,{\rm{K}}$, where grains are spun-up to suprathermal rotation and the ice mantles are disrupted due to centrifugal stress. The desorption size a_desp is smaller and a_desp,max is greater near the central star because of the higher radiation intensity and lower gas number density. In the surface layer where ${T}_{d}\gt 150\,{\rm{K}}$, icy grains of sizes larger than the grain core a_c = 0.05 μm are mostly desorbed. The intermediate layer where ${T}_{d}=30\mbox{--}150\,{\rm{K}}$ also witnesses great efficiency of rotational desorption, with a_desp drastically increasing from a_trans = 0.74 μm to ∼a_c = 0.05 μm as R declines and z increases. Interestingly, rotational desorption takes place in the region beyond the water snowline (around $3\,\mathrm{au}$ from the star) as well. Consequently, the ice mantles of dust grains in this region are already disrupted into smaller fragments, giving rise to the subsequent evaporation into water vapor. As a result, rotational desorption can destroy the ice mantles of grains beyond the snowline—and therefore push down the snowline in the vertical direction to the boundary beyond which the ice mantles start to be disrupted under the RATD effect.

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Furthermore, to demonstrate the effect of ro-thermal desorption on the release of COMs beyond the water snowline, we plot also in Figure 9 the region where ${\tau }_{\mathrm{sub},\mathrm{rot}}^{-1}({T}_{d})\,={\tau }_{\mathrm{sub},0}^{-1}({T}_{\mathrm{sub}})$ for different grain sizes a, where ${T}_{\mathrm{sub}}=150\,{\rm{K}}$ is taken to be the sublimation temperature and ${E}_{b}=5700\,{\rm{K}}$ the binding energy of water ice (Garrod 2013). In the surface layer, the snowlines in the cases with and without the rotation of grains become separated due to high T_d and low n_H increasing the rotation rate and thus decreasing the binding energy of molecules. As a result, the region that has a desorption rate equal to that of water ice by thermal desorption is extended further from the central star. The effect is more significant for larger grain sizes. Our model of ro-thermal desorption is not limited only to the water snowline. Plugging in the analogous parameters for a CO molecule, i.e., ${E}_{b}=1150\,{\rm{K}}$ (Garrod 2013) and ${T}_{\mathrm{sub}}=25\,{\rm{K}}$ (Mumma et al. 1993), one can see that the rotation of grains induced by RAT also increases the desorption rate of CO ice. Thus, the CO iceline under the framework of ro-thermal desorption would also be extended compared to the classical one. It is worth noting that the effect of rotational desorption is more efficient than that of ro-thermal desorption, i.e., rotational desorption can occur at more distant regions.

Figure 10 shows results similar to those of Figure 9, but for the disk around Herbig Ae/Be stars. The active region of rotational desorption is larger due to higher temperatures in the surface layer. Rotational desorption is also very efficient in the region beyond the snowline, which is much further from the star in this case (around $40\,\mathrm{au}$ from the central star). Up to $\sim 200\,\mathrm{au}$, desorption sizes still can reach $0.05\,\mu {\rm{m}}$ in the upper layer. At the outer radius $R=300\,\mathrm{au}$, the desorption of icy grains has not eased for high values of Σ₁ (e.g., 100 and $1000\,{\rm{g}}\,{\mathrm{cm}}^{-2}$) as in the case of T-Tauri disks. Thus, the snowline is pushed down further by rotational desorption in this case.

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In Figure 11, we plot the results for ro-thermal desorption with ${{\rm{\Sigma }}}_{1}=1\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ in regular R − z coordinates, for a better illustration of this effect on the snowline location.

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Figure 12 shows the desorption sizes of icy grains as a function of the disk height z at the different positions along the disk radius, with ${{\rm{\Sigma }}}_{1}=1\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. One can see that a_desp decreases exponentially with z to the lower boundary  0.05 μm, while a_desp,max increases exponentially until reaching the maximum value of ∼10 μm due to the decrease of n_H and the increase of T_d, similar to rotational disruption.

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To quantify the effect of rotational desorption on the presence of ice in the disks, we calculate the total surface mass density of water ice disrupted by rotational desorption

$$\begin{eqnarray}&&{\rm{\Delta }}{{\rm{\Sigma }}}_{\mathrm{ice},\mathrm{disr}}(R)={\int }_{-\infty }^{+\infty }{dz}{\int }_{{a}_{\mathrm{disr}}}^{{a}_{\mathrm{disr},\max }}{\rho }_{\mathrm{ice}}{V}_{\mathrm{ice}}(a)\displaystyle \frac{{{dn}}_{\mathrm{gr}}(a)}{{da}}{da},\end{eqnarray} \tag{ 28 }$$

where ${V}_{\mathrm{ice}}=4\pi ({a}^{3}-{a}_{{\rm{c}}}^{3})/3$ is the volume of the ice mantle of grain of size a with the core radius a_c, and the power-law distribution dn_gr(a)/da = Ca^−q is again assumed for icy grains. We consider a minimum grain size a_min = 0.05 μm and a maximum grain size a_max = 0.10 μm. The normalization coefficient C is calculated assuming the ratio of gas-to-dust mass to be 100. For thermal sublimation, we assume that the ice mantles of all grains with ${T}_{d}\gt 150\,{\rm{K}}$ are removed and compute the amount of sublimated ice ΔΣ_ice,sub. Results for ΔΣ_ice,disr and ΔΣ_ice,sub are shown in Figure 13. Prior to the snowline (at $R\sim 3\,\mathrm{au}$ for the T-Tauri disks and $R\sim 40\,\mathrm{au}$ for the Herbig Ae/Be disks), the amount of water ice sublimated through thermal sublimation is higher than that of ice disrupted by rotational desorption. This is expected, given that ice is believed to be present only in the region beyond the snowline where grain temperature is lower than the sublimation threshold ${T}_{d}\sim 150\,{\rm{K}}$ of H₂O. Therefore, according to our assumption, the mantles of all grains located before the snowline from the disk midplane to the surface are destroyed by thermal sublimation, whereas rotational does not occur in the disk interior and thus can affect a smaller surface mass density of water ice. However, beyond the snowline, rotational desorption continues to happen in the upper layer, and ice can still be removed in this region. The amount of disrupted ice is higher for smaller maximum grain sizes, for which more dust grains in the range size $[{a}_{\min },{a}_{\max }]$ are under the effect of RATD.

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4.2.2. Rotational Desorption Time versus Sublimation Time

Now let us compare the thermal sublimation time to the time it takes for rotational desorption to disrupt the ice mantles.

Following Hoang & Tram (2020), the sublimation time of the ice mantle of thickness Δa_m is given by

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{sub}}({T}_{d}) & = & -\displaystyle \frac{{\rm{\Delta }}{a}_{m}}{{da}/{dt}}=\displaystyle \frac{{\rm{\Delta }}{a}_{m}}{l\nu }\exp \left(\displaystyle \frac{{E}_{b}}{{T}_{d}}\right)\\ & \simeq & \ 1.5\times {10}^{3}\left(\displaystyle \frac{{\rm{\Delta }}{a}_{m}}{500\,\mathring{\rm A} }\right)\exp \left(\displaystyle \frac{{E}_{b}}{5700\,{\rm{K}}}\displaystyle \frac{100\,{\rm{K}}}{{T}_{d}}\right)\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 29 }$$

where da/dt ∼ l/τ_sub, with l being the thickness of ice monolayer, is the rate of decrease in the mantle thickness due to thermal sublimation.

We compute t_sub for desorption of water molecules with the binding energy ${E}_{b}=5700\,{\rm{K}}$, using the dust temperature from radmc-3d and the desorption time for the disks in the region where rotational desorption of ice mantles takes place from Equation (18), using numerical results for a_desp from Section 4.2. A comparison is provided in Figure 14 for grain size a = 1 μm and ${{\rm{\Sigma }}}_{1}=1\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. Rotational desorption of ice mantles is much faster than thermal sublimation in the surface and intermediate layers. The difference could be of order 10 in the inner central region.

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4.3.1. Ro-thermal Desorption of Molecules

We have seen that rotational desorption and ro-thermal desorption of ice mantles are both very efficient mechanisms to desorb COMs. As shown in Figures 9 and 10, for the grain model with a fixed core radius a_c = 0.05 μm and a varying mantle thickness, the region where rotational desorption is important appears to be larger than that of ro-thermal desorption. However, for a grain model of thin ice mantle, the resulting tensile stress by suprathermal rotation may be insufficient to overcome the adhesive energy between the mantle and the grain core surface. Therefore, it is hard to disrupt the entire ice mantle via RATD. In this case, the tensile strength is replaced by the adhesive strength, which depends on the mechanical properties of the surface and grain temperature. The adhesive strength is low for a clean surface, but it can reach $\sim {10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ for some rough surfaces (Work & Lian 2018). As shown in Hoang & Tung (2019), ro-thermal desorption of individual molecules can occur before the disruption of ice mantle if the ice mantle thickness is below 100 monolayers of water ice (i.e., Δa_m < 200 Å).

We thus consider a core-ice mantle model in which the mantle thickness Δa_m is fixed to a certain value while the core radius is varied. Results with ${{\rm{\Sigma }}}_{1}=1\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ are shown in 15. For Δa_m = 100 Å, the effect of rotational desorption of ice mantles becomes less significant than ro-thermal desorption for a = 0.5 μm in the case of T-Tauri disks and for a = 0.2 μm in the case of Herbig Ae/Be disks. For thinner mantles, the region where ice mantles are disrupted is narrower, and the active region of ro-thermal desorption is sufficiently broader for larger grain sizes. In that case, ro-thermal desorption takes over rotational desorption of ice mantles to release water and COMs into the gas phase. Furthermore, for this case of a thin mantle, the effect of rotational desorption is comparable to ro-thermal desorption, but less efficient than for the case of a thick ice mantle (see Figure 14).

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4.3.2. Ro-thermal Desorption of PAHs/Nanoparticles from the Ice Mantle

Ro-thermal desorption is found to be an efficient mechanism to desorb PAHs that are weakly bound to the ice mantle via the van der Waals force (Hoang & Tung 2019). Since ro-thermal desorption requires lower radiation strength to desorb than rotational desorption, one can describe the efficiency of ro-thermal desorption by considering the ejection threshold. Following Hoang & Tung (2019), one obtains the ejection threshold of PAHs:

$$\begin{eqnarray}&&{\omega }_{\mathrm{ej}}={\left(\displaystyle \frac{3{E}_{b}}{{{ma}}^{2}}\right)}^{1/2}\simeq \displaystyle \frac{{10}^{10}}{{a}_{-5}}{\left(\displaystyle \frac{({E}_{b}/k)}{4000{\rm{K}}}\displaystyle \frac{{m}_{{\rm{C}}6{\rm{H}}6}}{m}\right)}^{1/2}\mathrm{rad}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 30 }$$

where the respective binding energies of benzene C₆H₆ and naphthalene (C₁₀H₈) to ice are ${E}_{b}/k\sim 4000\,{\rm{K}}$ and $6000\,{\rm{K}}$ (see Tables 4 and 5 in Michoulier et al. 2018).

The ejection radiation strength is then

$$\begin{eqnarray}&&{U}_{\mathrm{ej}}\simeq 35{n}_{1}{T}_{2}^{1/2}(1+{F}_{\mathrm{IR}})\displaystyle \frac{{\lambda }_{0.5}^{1.7}}{\gamma {a}_{-5}^{1.7}}{\left(\displaystyle \frac{({E}_{b}/k)}{4000{\rm{K}}}\displaystyle \frac{{m}_{{\rm{C}}6{\rm{H}}6}}{m}\right)}^{1/2}\end{eqnarray} \tag{ 31 }$$

for a ≲ a_trans, and

$$\begin{eqnarray}&&{U}_{\mathrm{ej}}\simeq 1.8{n}_{1}{T}_{2}^{1/2}(1+{F}_{\mathrm{IR}})\displaystyle \frac{{\lambda }_{0.5}^{1.7}{a}_{-5}}{\gamma }{\left(\displaystyle \frac{({E}_{b}/k)}{4000{\rm{K}}}\displaystyle \frac{{m}_{{\rm{C}}6{\rm{H}}6}}{m}\right)}^{1/2}\end{eqnarray} \tag{ 32 }$$

for a > a_trans. Clearly, the ejection threshold is much lower than that of water and COMs. Therefore, the ro-thermal desorption is efficient for desorption of PAHs in star-forming regions.

It is difficult to explain the widespread presence of PAHs/nanoparticles in the surface layer of PPDs around Herbig Ae/Be stars, even in the inner gap (e.g., Boutéraon et al. 2019), because PAHs/nanoparticles in the surface layer are expected to be destroyed by EUV photons and the X-ray component of the star's radiation spectrum (Siebenmorgen & Krügel 2010). To explain the observations of PAHs in the disk, Siebenmorgen & Krügel (2010) suggested that PAHs are transported from the disk interior to the surface via turbulent mixing in the vertical direction, which requires the presence of PAHs in the disk interior. This scenario is difficult to reconcile with the fact that PAHs are expected to be depleted due to condensation into the ice mantle of dust grains in cold dense clouds (Bernstein et al. 1999; Cuylle et al. 2014; Cook et al. 2015). In this paper, we propose a top-down mechanism in which PAHs/nanoparticles are produced from the disruption of dust grains when they are being transported from the disk interior to the surface layers.

Our detailed modeling shows that large grains can be disrupted into nanoparticles in a vast area of the surface layer by the RATD mechanism. Moreover, PAHs/nanoparticles, which are frozen in the icy grain mantles, can desorb from the ice mantle via the ro-thermal desorption mechanism (Hoang & Tung 2019). Thus, if grains are small enough (a ≲ 1 μm) to be mixed with the gas by turbulence, they can escape grain settling and coagulation in the disk mid-plane and are transported to the surface layer (Dubrulle et al. 1995; Dullemond & Dominik 2004; Fromang & Nelson 2006, 2009). When exposed to stellar radiation, large grains within the range size [a_disr, a_disr,max] could be disrupted by RATD into PAHs/nanoparticles (see Figure 7). Furthermore, our results suggest an increase in the relative abundance of nanoparticles that corresponds with decreasing the distance to the star. Therefore, in the framework of rotational disruption and desorption, PAHs and nanoparticles would be continuously replenished in the surface layer, with the support of turbulent transport. This can resolve the longstanding mystery regarding the ubiquitous presence of PAHs/nanoparticles in the hot surface layer of PPDs.

Finally, spectroscopic and polarimetric measurements of the debris disk HD 32297 from Bhowmik et al. (2019) found an unusual blue NIR spectrum that corresponds to the presence of small grains of size much smaller than the bow-out limit due to radiation pressure (Wyatt et al. 1999). The authors explained this feature by a combination of several physical processes, including steady-state collisional evolution and the chain reaction of dust grain collisions triggered by the initial breakup of a large planetesimal near the central star, which Grigorieva et al. (2007) named the "avalanche" process. However, this process needs a two-belt structure for the breakups of massive planetesimals and the development and propagation of avalanche chain reaction (Thebault & Kral 2018).

Our numerical results for desorption sizes presented in Section 4.2 indicate that rotational desorption is very efficient in the disk surface. The ice mantles of icy grains of size in the range a_desp < a < a_desp,max are disrupted into small fragments. These tiny fragments can then be transiently heated to evaporation temperatures, inducing transient release of molecules, as theoretically demonstrated in Hoang & Tram (2020). Therefore, water ice and complex molecules frozen in the ice mantle can be released into the gas phase even at temperatures lower than their sublimation threshold. In particular, we predict that gas-phase water and COMs could be detected at locations well beyond the snowline of R > 3–100 au (see Figures 9 and 5).

To date, several complex molecules have been detected in PPDs, including CH₃OH and (HCOOH) in the TW Hydrae disk by Walsh et al. (2016) and Favre et al. (2018), and CH₃CN by Öberg et al. (2015). Recently, ALMA observations of the DG Tau disk by Podio et al. (2019) showed a lack of H₂CO emission in the warm inner disk, while the emission peak is located beyond the CO iceline ($\sim 40\,\mathrm{au}$). Interestingly, these molecules are detected far beyond the snowline at a distance above $10\,\mathrm{au}$. This cannot be explained by thermal sublimation, but rather is achieved via ro-thermal desorption as well as rotational desorption as shown in Figures 9 and 5.

We note that nonthermal desorption mechanisms such as cosmic-ray-induced desorption and UV photodesorption are not ruled out in PPDs (see, e.g., Öberg et al. 2015). Nevertheless, Walsh et al. (2010) carried out physical and chemical modeling of the PPD around the T-Tauri stars and found that cosmic-ray-induced desorption has little effect on the gas-phase abundances in the disk.

The water snowline characterized by the water sublimation temperature of ${T}_{d}\sim 150\,{\rm{K}}$ divides the inner region of rocky planets from the outer region of gas giant planets. Nevertheless, the precise location of the snowline is a longstanding problem in planetary science (Lecar et al. 2006; Min et al. 2011). In this paper, we studied the rotational desorption of ice mantles from the grain surface in PPDs.

Using the standard model of PPDs, we show that ice mantles from micron-sized grains (a < 100 μm) are disrupted and identify the location of snowline in the presence of rotational desorption. We find that the water snowline is extended outward. Figures 9 and 10 show that rotational desorption takes place in the region beyond the snowline—and thus can destroy the ice mantles of grains in this region. Indeed, observational detection of water ice from edge-on disks is reported in Terada et al. (2007) using water ice absorption features (Inoue et al. 2008). Future telescopes such as SPHEREx, GMT, the James Webb Space Telescope, SUBARU, and WFIRST would be useful to test the snowline. The rotational desorption of ice can have important implications for formation of planets, comets, and asteroids. Indeed, the total mass of rotationally disrupted ice is ${M}_{\mathrm{ice},\mathrm{disr}}$ = ${\int }_{{R}_{\mathrm{snowline}}}^{{R}_{\mathrm{out}}}2\pi r{\rm{\Delta }}{{\rm{\Sigma }}}_{\mathrm{ice},\mathrm{disr}}(r){dr}$ ∼ ${10}^{23}\mbox{--}{10}^{24}\,{\rm{g}}$ for T-Tauri disks and $\sim {10}^{25}\,{\rm{g}}$ for Herbig Ae/Be disks. As a result, the number of comets formed in the solar nebula would be significantly lower than previous estimates. On the other hand, the number of asteroids would be accordingly higher.

Various studies indicate that the C/Si abundance ratio in the Earth and rocky bodies in the inner solar system is several orders of magnitudes lower than the primordial interstellar ratio (see, e.g., Lee et al. 2010; Bergin et al. 2015; Anderson et al. 2017). Here, we suggest that rotational disruption can explain the carbon deficit if there exist two separate populations of carbonaceous and silicate grains in PPDs. Note that the two separate populations originate naturally from the rotational disruption of large aggregates by RATD if these original grains are a carbon-silicate mixture as expected from grain coagulation in dense regions.

Indeed, our present studies imply that carbonaceous nanoparticles can be reproduced by the disruption of large carbonaceous grains that are being transported from the disk interior to the surface layer (see Figures 5 and 6). When being exposed to harsh UV radiation, PAHs/carbonaceous nanoparticles are destroyed, converting solid carbon into the gas phase. The disruption of silicate grains by RATD is expected to be less efficient because of the effect of grain alignment. Indeed, silicate grains, which are paramagnetic material, are aligned by RATs along the ambient magnetic fields, a phenomenon referred to as B-RAT alignment; see Andersson et al. (2015) and Lazarian et al. (2015) for reviews. On the other hand, carbonaceous grains are diamagnetic material; as such, they are unlikely to be aligned with the magnetic field, rather than the radiation direction (Lazarian & Hoang 2007). If the stellar radiation makes an angle ψ with the ambient magnetic field, the rotation rate driven by RATs for silicate grains is smaller than that of carbon grains by a factor $\cos \psi$ (Hoang & Lazarian 2009, 2014) if the Larmor precession rate (${\tau }_{\mathrm{Lar}}^{-1}$) exceeds the precession rate around the radiation (τ_k⁻¹; Hoang & Lazarian 2016). The B-RAT alignment can occur closer to the star if silicate grains contain iron inclusions (Hoang & Lazarian 2016). The magnetic field geometry in PPDs is uncertain, but for an hourglass-shaped magnetic field, one expects that the stellar radiation forms a large angle with the magnetic field in the surface layers, i.e., $\cos \psi \lt 1$. As a result, RATD is, in general, more efficient for carbonaceous grains than for silicate grains, thus enhancing the conversion of large carbons into nanoparticles as well as the depletion of carbon depletion. A detailed study will be presented elsewhere.