Effect of Dust Rotational Disruption by Radiative Torques and Implications for F-corona decrease revealed by the Parker Solar Probe

A dust grain of radius $a$ rotating at velocity $\omega$ develops a centrifugal stress due to centrifugal force, which scales as $S=\rho a^{2}\omega^{2}/4$ with $\rho$ being the mass density of grain material (Hoang et al. 2019). When the rotation rate increases to a critical limit such that the tensile stress induced by centrifugal force exceeds the maximum tensile stress, the so-called tensile strength of the material ($S_{\max}$), the grain is disrupted instantaneously. The critical angular velocity for the disruption is given by

where $a_{-5}=a/(10^{-5}\,{\rm cm})$, $\hat{\rho}=\rho/(3\,{\rm g}\,{\rm cm}^{-3})$, $S_{\rm max}$ is the tensile strength of dust material and $S_{\rm max,9}=S_{\rm max}/(10^{9}\,{\rm erg}\,{\rm cm}^{-3})$ is the tensile strength in units of $10^{9}\,{\rm erg}\,{\rm cm}^{-3}$.

The exact value of $S_{\max}$ depends on the dust grain composition and structure. Compact grains have higher $S_{\max}$ than porous/composite grains. Ideal material without impurity, such as diamond, can have $S_{\max}\geq 10^{11}\,{\rm erg}\,{\rm cm}^{-3}$ (Burke & Silk 1974; see Hoang et al. 2019 for more details), while dust aggregates have lower tensile strength (see Tatsuuma et al. 2019; Kimura et al. 2020 for more details). In the following, we consider a range of the tensile strength from $S_{\max}=10^{8}-10^{11}\,{\rm erg}\,{\rm cm}^{-3}$. The largest $S_{\max}$ is expected for nanoparticles that are likely to have compact structures.

Let $u_{\lambda}$ be the spectral energy density of radiation field at wavelength $\lambda$. To describe the strength of a radiation field, let define $U=u_{\rm rad}/u_{\rm ISRF}$ with $u_{\rm ISRF}=8.64\times 10^{-13}\,{\rm erg}\,{\rm cm}^{-3}$ being the energy density of the average interstellar radiation field (ISRF) in the solar neighborhood as given by Mathis et al. (1983).

The radiation energy density at distance $R$ from the Sun is given by

where $L_{\star}=L_{\odot}$ for the solar bolometric luminosity.

The radiation strength at distance $R$ becomes

where $U_{0}=5.2\times 10^{7}$ is the radiation strength at $R_{0}=1\,{\rm AU}$.

The mean wavelength of the solar radiation field is defined as

which yields $\bar{\lambda}\approx 0.91\,{\mu\rm{m}}$ for the solar-type star.

Grains subject to the anisotropic radiation field experience RATs which act to spin-up the grains to suprathermal rotation (Dolginov & Mitrofanov 1976; Draine & Weingartner 1996; Lazarian & Hoang 2007a; Hoang & Lazarian 2008). At the same time, grain rotation experiences damping due to collisions with gas species (of proton density $n_{{\rm H}}$ and temperature $T_{\rm gas}$) and infrared emission (see Appendix C). The grain rotation velocity induced by RATs is given by (Hoang et al. 2019 and Hoang 2019)

for grains with $a\lesssim a_{\rm trans}$, and

for grains with $a>a_{\rm trans}$ where $a_{\rm trans}\sim\bar{\lambda}/1.8$ is the transition grain size where RAT efficiency $Q_{\Gamma}$ changes the slope from $\eta\sim-3$ to $\eta=0$ (Lazarian & Hoang 2007a; see Appendix C).

In the above equations, $\overline{\lambda}_{0.5}=\overline{\lambda}/0.5\,{\mu\rm{m}}$, $\hat{n}=n_{H}/10\,{\rm cm}^{-3},\hat{T}_{\rm gas}=T_{\rm gas}/100\,{\rm K}$, $F_{\rm IR}$ is the dimensionless parameter that describes the grain rotational damping by infrared emission. In addition to gas and IR damping, grains experience rotational damping due to plasma drag and ion collisions (Draine & Lazarian 1998; Hoang et al. 2010). However, subject to intense solar radiation field, the infrared damping is dominant ($F_{\rm IR}\gg 1$), such that the grain rotation rate does not depend on the local gas properties (see Appendix C for more details).

For strong solar radiation fields considered in this paper, $F_{\rm IR}\gg 1$, from Equations (5) and (1), one can obtain the disruption grain size:

for $a_{\rm disr}\leq a_{\rm trans}$.

At $R=1\,{\rm AU}$, one obtains $a_{\rm disr}\sim 0.02\,{\mu\rm{m}}$ for the typical parameters of $S_{\max,9}=1$, $\gamma=1$, and $\lambda=0.91\,{\mu\rm{m}}$. At $R=0.1\,{\rm AU}$ ($21R_{\odot}$, one has $a_{\rm disr}=0.011\,{\mu\rm{m}}$. At $R=0.05\,{\rm AU}$ ($10R_{\odot}$), $a_{\rm disr}\sim 0.0096\,{\mu\rm{m}}$. At $R=0.02\,{\rm AU}~{}(4R_{\odot}$), $a_{\rm disr}=0.0076\,{\mu\rm{m}}$. So, within $R=0.5\,{\rm AU}$, nanoparticles of size $a<a_{\rm disr}/2\sim 0.0055\,{\mu\rm{m}}$ (5.5 nm) are abundant due to RATD.

Due to the decrease of $\omega_{\rm RAT}$ with the grain size for $a>a_{\rm trans}$ (see Eq. 5), very large grains would not be disrupted. The maximum size of grains that can still be disrupted by RATD is given by (Hoang 2019)

Figure 1 (upper panel) shows the grain disruption size vs. the heliocentric distance for the different tensile strength. For $R<5R_{\odot}$ (0.24 AU), only very small grains of sizes $a<0.005\,{\mu\rm{m}}$ (5 nm) can survive against RATD, assuming compact grains of $S_{\rm max}\sim 10^{9}\,{\rm erg}\,{\rm cm}^{-3}$. Ideal nanoparticles with $S_{\rm max}\sim 10^{11}\,{\rm erg}\,{\rm cm}^{-3}$ can have a maximum size of $a<0.03\,{\mu\rm{m}}$ (30 nm). For $R<50R_{\odot}$ (0.5 AU), we predict only small grains of $a<0.025\,{\mu\rm{m}}$ (25 nm).

The characteristic timescale for rotational desorption can be estimated as (Hoang et al. 2019):

for $a_{\rm disr}<a\lesssim a_{\rm trans}$, and

for $a_{\rm trans}<a<a_{\rm disr,max}$ where $U_{7}=U/10^{7}$.

Figure 1 (lower panel) shows the disruption time obtained for the different tensile strengths. The disruption time decreases rapidly with the heliocentric distance.

Our results reveal that grains within $0.1$ AU ($21R_{\odot}$) are dominated by nanoparticles of size $a<0.01\,{\mu\rm{m}}$ (10 nm) due to RATD, as illustrated in Figure 2).

The solar wind usually consists of very slow ($v\sim 200-300\,{\rm km}\,{\rm s}^{-1}$), slow ($v\sim 300-700\,{\rm km}\,{\rm s}^{-1}$), and fast ($v>700\,{\rm km}\,{\rm s}^{-1}$) solar winds (see, e.g., Mukai & Schwehm (1981)). The very slow winds (VSLW) are observed frequently in the inner solar region of heliocentric distance $R<1\,{\rm AU}$ such as by Helios 1/2, but are rarely near the Earth and the density of VSLW is much higher than that of fast solar winds (Sanchez-Diaz et al. 2016).

The number density of solar wind protons varies with the heliocentric distance and is given by

where $n_{0}=6\,{\rm cm}^{-3}$ at $R_{0}=1\,{\rm AU}$ (see e.g., Venzmer & Bothmer (2018)). At heliodistance of $R=0.1\,{\rm AU}$ ($\sim 21R_{\odot}$), the proton density increases to $n_{p}=600\,{\rm cm}^{-3}$.

Bombardment of energetic protons gradually erodes the grain surface via the sputtering mechanism (see e.g., Hoang et al. 2015; Hoang & Lee 2020).

To calculate the sputtering time, we assume a uniform distribution of proton velocity, and the mean velocity of solar wind (sw) protons is $\bar{v}_{\rm sw}\sim 400\,{\rm km}\,{\rm s}^{-1}$. Let $Y_{\rm sp}$ be the sputtering yield induced by the proton bombardment. The characteristic timescale, $t_{\rm nsp}$, of grain destruction by nonthermal sputtering for a spherical grain of size $a$ is defined by

where $n_{2}=n_{\rm p}/(100\,{\rm cm}^{-3})$, and $\bar{A}_{sp}$ is the mean atomic mass of sputtered atoms from the grain (see e.g., Hoang & Lee 2020).

At $R\sim 0.1\,{\rm AU}$ ($\sim 21R_{\odot}$), $n_{p}\sim 600\,{\rm cm}^{-3}$ (see Eq. 11), the sputtering time is $t_{\rm nsp}=1093(a/10^{-7}\,{\rm cm})$ days, assuming $Y_{\rm sp}=0.5$. Thus, the sputtering time is much longer than the rotational disruption time $t_{\rm disr}$ for grains of $a>a_{\rm dirs}$ (see the right panel of Figure 1).

To study if nonthermal sputtering can be fast enough to destroy nanoparticles before they are dragged to the sublimation zone by the Poynting-Robertson (P-R) drag, we calculate the ratio of nonthermal sputtering time to the P-R drag time (see Eq. B3):

where $\langle Q_{\rm pr}\rangle$ is the averaged radiation pressure cross-section efficiency which is a function of the grain size.

For nanoparticles in the solar radiation field with $\langle Q_{\rm pr}\rangle\lesssim 0.1$ (see Figure 7), Equation (13) reveals that nanoparticles are rapidly destroyed by nonthermal sputtering before dragged into the sublimation zone by the P-R drag. The efficiency of nonthermal sputtering is increased with decreasing the heliocentric distance due to the increase of proton density (see Equation 11).

We note that for nanoparticles, rotational disruption by stochastic mechanical torques by the solar wind is less efficient than nonthermal sputtering because energetic protons tend to pass through the grain (Hoang & Lee 2020). Misconi (1993) studied the rotational bursting of circumsolar dust caused by solar winds protons using the ”Paddack effect”–”windmill” (Paddack & Rhee 1975). However, Misconi (1993) did not take into account the effect of proton passage and assumed a rather low tensile strength of $S_{\max}=10^{6}\,{\rm erg}\,{\rm cm}^{-3}$ for circumsolar dust, which is quite low for dust grains of compact structures. The rotational disruption by regular mechanical torques due to the interaction of the solar wind with irregular grains (Lazarian & Hoang 2007b; Hoang et al. 2018) is expected to be more efficient. A detailed study of this effect is presented elsewhere.

Figure 3 compares the nonthermal sputtering time evaluated at $a=a_{\rm disr}$ and $a=a_{\rm disr}/2$ to other relevant timescales. The sputtering yield for protons of $\bar{v}_{\rm sw}$ is calculated as in Hoang & Lee (2020). The P-R drag time is calculated using Equation (B3) where $\langle Q_{\rm pr}\rangle$ is a function of the grain size as shown in Figure 7. The sublimation time $t_{\rm sub}$ for two grain sizes at $T_{d}=1500\,{\rm K}$ is shown for comparison (see Eq. B5).

The dashed vertical lines denote the location of the extended dust-free-zone determined by nonthermal sputtering $R_{\rm sp}$. The sputtering radius is $R_{\rm sp}\sim 7-13R_{\odot}$ for grains of $S_{\rm max}\sim 10^{9}-10^{11}\,{\rm erg}\,{\rm cm}^{-3}$ for silicate and $8R_{\odot}$ for $S_{\rm max}\sim 10^{9}\,{\rm erg}\,{\rm cm}^{-3}$ for graphite. Note that graphite is not found in the interplanetary dust, the choice of graphite here is an example of a highly absorptive dust, such as hydrogenated amorphous carbon.

In Figure 3, the region where $t_{\rm nsp}(a=a_{\rm disr})\lesssim t_{\rm sub}(T_{d}=1500\,{\rm K})$ is considered an extended dust-free-zone. Note that the classical dust-free-zone is defined by thermal sublimation where grains are destroyed rapidly on a timescale of $t_{\rm sub}(T_{d}=1500\,{\rm K})$. Thus, if nonthermal sputtering could destroy grains in the same time beyond this zone, then, the dust-free-zone is extended.

Beyond the extended dust-free-zone, nanoparticles of $a<a_{\rm disr}$ can be depleted by nonthermal sputtering because $t_{\rm nsp}(a)<t_{\rm sub}$. As a result, the mass loss of dust via nonthermal sputtering is decreasing outward, or the total dust mass decreases inward until the dust-free-zone.

Let $a_{\rm nsp}$ be the critical size of nanoparticles that are destroyed by nonthermal sputtering. One can determine $a_{\rm nsp}$ by setting $t_{\rm nsp}(a_{\rm nsp})=t_{\rm sub}$. All grains smaller than $a_{\rm nsp}$ are destroyed by nonthermal sputtering. The value of $a_{\rm nsp}$ will increase with decreasing $R$ due to the increase of the proton density (see Eq. 11).

Figure 4 illustrates the grain size distribution ($dn/da\propto a^{-3.5}$) as a result of the RATD mechanism and nonthermal sputtering. Only small grains within a narrow range (shaded area) could survive in the F-corona.

Let $f_{\rm high-J}$ be the fraction of grains that can be spun-up to suprathermal rotation and then be disrupted by RATD (see more in Section 4. Then, we can calculate the decrease of the dust mass as a function of heliocentric distance as follows:

where $M_{d}(R,0)$ is the original dust mass in the absence of sputtering

where a power-law grain size distribution of $dn/da=Ca^{-3.5}$ with $a_{\rm min}=3$ Å  is adopted.

Figure 5 shows the variation of the relative dust mass $M_{d}/M_{d}(0)$ vs. the heliocentric distance as a result of RATD and sputtering for the different tensile strength and perfect disruption with $f_{\rm high-J}=1$. We see that $M_{d}/M_{d}(0)$ starts to decrease considerably from $R\sim 0.2$ AU ($42R_{\odot}$), and a significant mass loss occurs at $R\lesssim 0.03\,{\rm AU}~{}(6R_{\odot})$, which suggests an extended dust-free-zone. Grains made of weak material are more efficiently disrupted by RATD and experience larger mass loss.

Note that in the absence of RATD, nonthermal sputtering still can remove nanoparticles, but the fraction of depleted mass is negligible because the dust mass is mostly dominated by largest grains of $a\sim a_{\rm max}$. The effect of RATD converts large grains $a>a_{\rm disr}$ into smaller sizes and increases the abundance of nanoparticles. As a result, the amount of dust mass depleted by sputtering is significantly enhanced. However, it is uncertain whether RATD is efficient for very large grains (VLGs) of $a>\bar{\lambda}/0.1\sim 9\,{\mu\rm{m}}$ due to the lack of numerical calculations of RATs. If circumsolar dust is dominated by VLGs, the effect of RATD and nonthermal sputtering is less efficient.

We have shown that RATD rapidly breaks large dust grains into nanoparticles. This results in the increase in the abundance of small grains relative to large ones toward the Sun. Subsequently, nonthermal sputtering by the solar wind destroys smallest grains. Due to the increase of the solar wind density with decreasing the heliocentric distance, nonthermal sputtering rate is rapidly increased. As a result, dust grains can be completely removed by nonthermal sputtering near the Sun, which extends the dust-free-zone implied by thermal sublimation. We found that the extended dust-free-zone is located within a radius of $R_{dfz}\sim 6-12R_{\odot}$ (depending on $S_{\rm max}$, which is much larger than the classical one defined by thermal sublimation of $4-5R_{\odot}$.

Our results also suggest that F-corona dust as well as dust in the inner solar system ($R<1\,{\rm AU}$) mostly contain nanoparticles of size $a\sim 1-10$ nm (see Figure 4). This is a plausible explanation for the existence of nanodust detected by in-situ measurements (see, e.g., Mann et al. 2007; Mann 2017; Ip et al. 2019). This RATD mechanism is more efficient than collisional fragmentation previously thought (see e.g., Mann et al. 2007 for discussion of various mechanisms to form nanodust in the inner solar system).

In-situ measurements by dust detector on board the Helio spacecrafts reported the F-corona decrease at heliocentric distances between $D=0.3-1$ AU (Gruen et al. 1985). This is thought due to the mutual collisions that makes grains smaller and decrease of the forward scattering cross-section. However, the RATD appears to be more efficient in producing small grains due to its short timescale.

Note that RATD is valid for grains of $\bar{\lambda}/a>0.1$. For larger grains of $a>\bar{\lambda}/0.1\sim 9\,{\mu\rm{m}}$, RATs of such very large grains are not yet available due to the lack of numerical calculations because it requires expensive computations to achieve reliable results of RATs for grains of $2\pi~{}a/\lambda\gg 1$ (see e.g., Draine & Flatau 2004). Expecting the decrease of RATs with increasing $a$, the disruption may still be important when the decrease of RATs is compensated by the increase of the radiation energy density.

The first-year results from the PSP at heliocentric distances of $D=0.16-0.25$ AU ($34.3-53.7R_{\odot}$) reveal the gradual decrease the F-corona (Howard et al., 2019). With the elongation $\epsilon\sim 15-20^{\circ}$, one can estimate the corresponding elongation in solar radii $R_{\epsilon}\sim(0.166-0.336)\sin(15^{\circ})\,{\rm AU}\sim 9-19R_{\odot}$.

To explain the first-year PSP results, let us first recall that the intensity of scattered sunlight by dust depends on the dust scattering cross-section as follows:

where the term in the second bracket describes the dust mass.

Equation (16) reveals that the decrease of dust mass by sputtering and RATD as shown in Figure 5 directly reduces the observed intensity. Moreover, the decrease of grain sizes by RATD also decreases the scattering efficiency $Q_{\rm sca}$ because in the small grain limit ($a\ll\lambda$), $Q_{\rm sca}(a,\lambda)\sim(2\pi a/\lambda)^{4}$. Thus, the joint effect of RATD and sputtering would decrease the brightness of the F-corona.

Figure 6 illustrates the F-corona as a result of RATD and nonthermal sputtering, which would be observed with the PSP. The extended dust-free-zone is located at $R=6R_{\odot}$. Beyond this radius, the F-corona decreases with the radius, starting from $42R_{\odot}$ to the dust-free-zone.

Note that the sublimation radius given by Equation (B10) reveals the range of the dust-free-zone between $4-5R_{\odot}$ for silicate grains of sizes $a\sim 0.1-0.001\,{\mu\rm{m}}$. Therefore, even with the effect of RATD, thermal sublimation alone cannot explain the thinning-out of circumsolar dust observed from $R_{\epsilon}\lesssim 19R_{\odot}$. However, our results shown in Figure 5) indicate that the joint effect of RATD and nonthermal sputtering could successfully explain the gradual decrease of F-corona toward the Sun. Moreover, the PSP’s observation is consistent with our result with the largest value of $S_{\max}$ because this model predicts the F-corona decrease from a closest distance of $19R_{\odot}$.

The PSP is planned to undergo 24 orbits around the Sun. The latest orbits (22-24) will reach the closest distance of $10.74R_{\odot}$ (Fox et al. 2016). Previous studies predict that the dust-free-zone is between $4-5R_{\odot}$, which cannot be confirmed with the PSP. We found that the joint action of RATD and sputtering increase the radius of dust-free-zone to $7-11R_{\odot}$ (see Figure 3). This would be tested with the upcoming orbits of the PSP.

Finally, it is worth to mention that the variation of the grain size distribution by RATD affects the effect of radiation pressure. Indeed, as grains become smaller, the radiation cross-section efficiency $\langle Q_{\rm pr}\rangle$ decreases with decreasing grain size (see Figure 7). Thus, the ratio of radiation force to gravity $\beta=F_{\rm rad}/F_{\rm gra}$ becomes smaller than unity (see Equation B2), such that radiation pressure cannot blow these nanoparticles away.

We have assumed that RATs spin up grains to their maximum rotational angular velocity and ignored the effect of grain alignment. Thus, our obtained results (e.g., Figure 5) are considered an upper limit of the rotational disruption effect. Previous studies show that in addition to spin up, RATs induce grain alignment along the magnetic field direction, which is usually referred to as B-RAT. Moreover, subject to the intense solar radiation, the alignment axis may change from the magnetic field (B-RAT) to the radiation direction (i.e., k-RAT; see more details in Lazarian & Hoang 2019). As suggested by Parker (1958), the magnetic field near the Sun is almost radial. Therefore, our obtained results are weakly affected by the magnetic field because the radiation and magnetic field are nearly parallel.

We have also assumed that all aligned grains have their maximum angular momentum $\omega_{\rm RAT}$ (Eqs. 5 and (C11), which corresponds to the situation that all grains are driven to high-J attractors (Hoang & Lazarian 2016; Hoang & Lazarian 2008). In general, the fraction of grains on high-J attractors, denoted by $f_{\rm high-J}$, depends on the grain properties (shape, size, and magnetic properties), and $0<f_{\rm high-J}\leq 1$. An extensive study for numerous grain shapes and compositions by Herranen et al. (2021) found $f_{\rm high-J}\sim 0.2-0.7$, depending on the grain shapes. The presence of iron inclusions is found to increase $f_{\rm high-J}$ to unity (Hoang & Lazarian, 2016). It is likely that for a given size, the grains with iron inclusions would be fully disrupted first. As a consequence, we may expect to have an increase of pure “iron” tiny grains or grains with significantly increased magnetic content and pure paramagnetic grains. For the latter fraction, thermal sublimation will be only the destruction effect for the fraction of grains without high-J attractors that RATD is not effective. At the same time, our predicted dependence will be applicable to the population of high-J grains. More details about the relation of alignment and the RATD are given in Lazarian & Hoang (2021). Therefore, the decrease of dust mass predicted in this paper (Fig. 5) is an upper limit and may be lower by a factor $f_{\rm high-J}$. As a result, the decrease of $M_{d}$ (F-corona) would start from a smaller heliodistance of $<42R_{\odot}$.

Circumsolar dust is an ideal environment to test RATD mechanism because we have both indirect and direct measurement of dust. Observations at ultraviolet, visible, and near-infrared wavelength reveal the existence of large grains between $1-10\,{\mu\rm{m}}$ in the circumsolar corona around $4-5R_{\odot}$. This implies that the RATD effect is not perfect (i.e., $f_{\rm high-J}<1$), as we discussed in the previous subsection. Detailed modeling of the F-corona brightness with observations would provide a constraint on the efficiency of RATD.