4 Energy balance of dust grains

The surface chemical reactions responsible for the growth of a dust particle liberate the latent heat of condensation $\Delta_{\rm f}\hspace{-0.2ex}H$ [erg/g] which causes a heating of the grain as

$$Q_{\rm cond} = \frac{{\rm d}V}{{\rm d}t}~\rho_{\rm d}~\Delta_{\rm f}\hspace{-0.2ex}H.$$ (39)

It has been proposed (Cooper ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$2002) that this heating can increase the internal dust temperature $T_{\rm d}$ substantially, until the sublimation temperature is reached (where $S\!=\!1$). In that case, the growth rate is limited by the need to remove the latent heat of condensation from the grain and Eqs. (22) and (32) are not valid.

A further heating process of the dust particle is given by the friction caused by the motion relative to the gas,

$$Q_{\rm fric} ~=~ \alpha_{\rm fric}~\vert\vec{F}_{\rm fric}\!\... ...; \vec{g}\!\cdot\!\vec{v}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}.$$ (40)

Hereby, we assume that the total work done by the frictional force per time, $\vert\vec{F}_{\rm fric}\!\cdot\!\vec{v}_{\rm dr}\vert$, is converted into heat, from which the fraction $\alpha_{\rm fric}$ is delivered to the grain. For the r.h.s. expression, the condition of equilibrium drift $\vert\vec{F}_{\rm fric}\vert\!=\!\vert\vec{F}_{\rm grav}\vert$ is used (Eq. (5)). Since elastic collisions do not transfer any energy, we assume $\alpha_{\rm fric}\!=\!\alpha_{\rm acc}$ (see Eq. (43)).

In order to determine the dust temperature increase, we balance these heating processes with the net energy losses due to radiation and due to inelastic collisions. The net radiative cooling rate of a single dust grain is given by

$$Q_{\rm rad} = 4\pi\!\!\int\!\!\pi a^2 Q_{\rm abs}(a,\lambda) ... ...ig[B_{\lambda}(T_{\rm d})-J_{\lambda}\big] ~{\rm d}\lambda \ ,$$ (41)

where $Q_{\rm abs}$ is the absorption efficiency, $B_{\lambda}$ the Planck function and $J_{\lambda}$ the mean intensity of the radiation field.

The cooling due to inelastic collisions with gas particles, in particular with H₂, depends again on the Knudsen number. For large Knudsen numbers ( ${Kn}\!\gg\!1$) the collisional cooling rate is given by

$$Q_{\rm coll}^{\rm lKn} = \pi a^2~n~\bar{v}_{\rm th}~ \alpha_{\rm acc}~2k(T_{\rm d}-T_{\rm g}) ,$$ (42)

where $T_{\rm g}$ is the gas temperature, $\bar{v}_{\rm th}$the mean thermal velocity introduced on page 300 and $\alpha_{\rm acc}$ the efficiency for thermal accommodation, given by Burke ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Hollenbach (1983)

$$\alpha_{\rm acc} = 0.1+0.35\cdot\exp \left(-\sqrt{\frac{T_{\rm d}+T_{\rm g}}{500~{\rm K}}}\right) \cdot$$ (43)

For small Knudsen numbers ( ${Kn}\!\ll\!1$), the removal of heat from the grain's surface in the laminar case is limited by the heat conductivity of the ambient gas. We consider the energy equation of an ideal fluid

$$\frac{{\rm\partial} }{{\rm\partial} t}(\rho~e) + \nabla \big(\vec{v}_{\rm gas}[\rho~e+P]\big) = - \nabla \vec{j}_{\rm HC}$$ (44)

where e is the internal energy of the gas including kinetic and gravitational potential energies (Paper I) and P the thermal gas pressure. The energy flux by heat conduction is given by

$$\vec{j}_{\rm HC}= - \kappa\nabla T .$$ (45)

$\kappa\;\rm [erg~K^{-1}cm^{-1}s^{-1}]$ is the heat conductivity of the gas which according to Jeans (1967) equals

$$\kappa = {\textstyle\frac{9\gamma-5}{4}}~\mu_{\rm kin}~C_{\rm V} ,$$ (46)

where $\gamma\!=\!(f\!+\!2)/f\!=\!7/5$ is the adiabatic index, f the number of degrees of freedom, $\mu_{\rm kin}$ the kinematic viscosity (Eq. (9)) and $C_{\rm V}\!=\!(fk)/(2\bar{\mu})\approx 8.845\times 10^7~\rm erg~g^{-1}~K^{-1}$the isochoric heat capacity of the gas, resulting in

$$\kappa = 988~{\rm\frac{erg}{\rm K~cm~s}}~\sqrt{T_{\rm g}\rm [K]} .$$ (47)

The further derivation of the cooling rate by heat conduction is analogous to the derivation of the viscous growth rate on page 303. We consider the static case ( $\vec{v}_{\rm gas}\!=\!0$) and assume stationary ( $\partial(\rho e)/\partial t\!=\!0$) and spherical symmetry, such that Eq. (44) becomes

$$r^2\kappa~\frac{{\rm\partial} T}{{\rm\partial} r} = {\rm const} .$$ (48)

Regarding $\kappa$ as a constant, the solution of Eq. (48) with boundary conditions $T(a)\!=\!T_{\rm d}$ and $T(\infty)\!=\!T_{\rm g}$ is

$$T(r) = T_{\rm g}-\frac{a}{r}\big(T_{\rm g}-T_{\rm d}\big)$$ (49)

and the collisional cooling rate $Q_{\rm coll}^{\rm sKn}\!=\!4\pi r^2 \vec{j}_{\rm HC}$, according to Eqs. (45) and (49), results in

$$Q_{\rm coll}^{\rm ~sKn} = 4\pi~\kappa~a~(T_{\rm d}-T_{\rm g}) \ .$$ (50)

The general collisional cooling rate with $Kn'\!=\!Kn/Kn^{\rm cr}$ is again approximated by

$$Q_{\rm coll} ~=~ Q_{\rm coll}^{\rm lKn} \!\left(\frac{Kn'}{... ...Q_{\rm coll}^{\rm sKn} \!\left(\frac{1}{Kn'+1}\right)^{\!2}$$ (51)

where $Kn^{\rm cr}$ results from equating Eqs. (42) with (50). We find a value of 0.023 at 1000 K and a value of 0.019 at 2000 K. For simplicity, we apply a constant value for the critical Knudsen number as $Kn^{\rm cr} = 0.02$ in the following.

The energy balance of a single dust grain is finally given by

$$Q_{\rm cond} + Q_{\rm fric} ~=~ Q_{\rm rad} + Q_{\rm coll} ,$$ (52)

which states an implicit equation for the temperature increase $\Delta T=T_{\rm d}-T_{\rm g}$ due to the liberation of latent heat during grain growth and frictional heating.

Figure 4: Contour plot of the temperature increase $\log \Delta T$ [K], due to the liberation of latent heat during grain growth and frictional heating, as function of the grain radius a and gas density $\rho$ at constant gas temperature $T\!=\!1500$ K for quartz grains with the same parameters as in Fig. 3. We assume growth by accretion of the key species SiO with maximum particle density $n_{\rm SiO} = 10^{(7.55-12)} n_{\rm \langle H\rangle}$ and extreme supersaturation ( $S\!\to\!\infty$). The two dashed lines indicate where the two considered heating and cooling rates are equal. Above these lines, $Q_{\rm fric}\!>\!Q_{\rm cond}$ and $Q_{\rm rad}\!>\!Q_{\rm coll}$, respectively.

Figure 4 shows an example for the resulting temperature increase $\Delta T$ of quartz grains. We assume $J_{\lambda}\!=\!B_{\lambda}(T_{\rm g})$ and again consider the explicit growth reaction (Eq. (38)) with a release of latent heat of $\Delta_{\rm f}\hspace{-0.2ex}H\!=\!5.73~$eV per reaction at 1000 K, 5.61 eV at 1500 K and 5.49 eV at 2000 K (data reduced from the enthalpies of formation of the involved molecules and the solid, source: JANAF-tables, electronic version, Chase ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1985).

For the sake of simplicity, we furthermore assume

$$Q_{\rm abs}(a,\lambda) \approx \min\left\{1~,~ a~\widehat{Q}^{\rm ~SPL}_{\rm ext}(\lambda)\right\}$$ (53)

for this calculation, where the extinction efficiency over a in the small particle limit of Mie theory is given by

$$\widehat{Q}^{\rm ~SPL}_{\rm ext} = \frac{8\pi}{\lambda}~ {\c... ...\!\left\{\frac{m(\lambda)^2-1} {m(\lambda)^2+2}\right\} \cdot$$ (54)

$m(\lambda)$ is the complex refractory index of the dust grain material and ${\cal I}m$ the imaginary part. The optical constants for amorphous SiO₂ (quartz glass) are taken from H. R. Philipp's section in (Palik 1985) and the resulting $\widehat{Q}^{\rm ~SPL}_{\rm ext}$-values are log-log-interpolated between the measured wavelengths points.

Despite these simplifications, Fig. 4 clearly indicates that the warming of the dust grains due to the release of latent heat is negligible, being less than 3.5 K all over the relevant parts of the size-density-plane, where $\tau_{\rm gr}\!\leq\!\tau_{\rm sink}$(compare Fig. 3). Here, we find that this heating is balanced by collisional cooling $Q_{\rm cond}\!\approx\!Q_{\rm coll}$. Since both heating/cooling rates scale as $\propto\!a^2\rho$ for large Knudsen numbers and as $\propto\!a$ for small Knudsen numbers, a constant value for $\Delta T$ tunes in for both cases, $\Delta T\!\approx\!3.5$ K for large Kn, and $\Delta T\!\approx\!0.5$ K for small Kn. Note that the calculated temperature differences are always an upper estimate. The actual temperature differences may be much smaller because we have assumed solar, undepleted abundances of Si in the gas phase and $S\!\to\!\infty$ for the calculation of ${\rm d}V/{\rm d}t$.

For larger particles (roughly at $a\!\ga\!a_{\rm max}$ as defined in Sect. 3.1), the character of the energy balance of the dust particles changes. Here, the frictional heating due to the rapid relative motion and the radiative cooling dominate, i.e. $Q_{\rm fric}\!\approx\!Q_{\rm rad}$. Much larger temperature deviations up to 10 000 K result in this case. However, as argued before, such large grains cannot be formed in brown dwarf atmospheres.

Thus, the resulting increase of the dust temperature $T_{\rm d}$ is by far too small to reach the sublimation temperature $T_{\rm sub}$, unless gas temperatures very close to $T_{\rm sub}$ are considered and, therefore, Eqs. (22) and (32) remain valid.