4 Energy balance of dust grains

The surface chemical reactions responsible for the growth of a dust
particle liberate the latent heat of condensation
[erg/g] which
causes a heating of the grain as

(39) |

It has been proposed (Cooper 2002) that this heating can increase the internal dust temperature substantially, until the sublimation temperature is reached (where ). 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,

Hereby, we assume that the total work done by the frictional force per time, , is converted into heat, from which the fraction is delivered to the grain. For the r.h.s. expression, the condition of equilibrium drift is used (Eq. (5)). Since elastic collisions do not transfer any energy, we assume (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

where is the absorption efficiency, the Planck function and the mean intensity of the radiation field.

The cooling due to inelastic collisions with gas particles, in particular with
H_{2}, depends again on the Knudsen number. For large Knudsen numbers (
)
the collisional cooling rate is given by

where is the gas temperature, the mean thermal velocity introduced on page 300 and the efficiency for thermal accommodation, given by Burke Hollenbach (1983)

For small Knudsen numbers ( ), the removal of heat from the grain's surface in the laminar case

where

is the heat conductivity of the gas which according to Jeans (1967) equals

(46) |

where is the adiabatic index,

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 ( ) and assume stationary ( ) and spherical symmetry, such that Eq. (44) becomes

Regarding as a constant, the solution of Eq. (48) with boundary conditions and is

and the collisional cooling rate , according to Eqs. (45) and (49), results in

The general collisional cooling rate with is again approximated by

where 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 in the following.

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

(52) |

which states an implicit equation for the temperature increase due to the liberation of latent heat during grain growth and frictional heating.

Figure 4:
Contour plot of the temperature increase
[K],
due to the liberation of latent heat during grain growth and
frictional heating, as function of the grain radius a and gas
density
at constant gas temperature
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
and extreme
supersaturation (
). The two dashed lines indicate
where the two considered heating and cooling rates are equal.
Above these lines,
and
,
respectively. |

Figure 4 shows an example for the resulting temperature increase of quartz grains. We assume and again consider the explicit growth reaction (Eq. (38)) with a release of latent heat of 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 1985).

For the sake of simplicity, we furthermore assume

(53) |

for this calculation, where the extinction efficiency over

(54) |

is the complex refractory index of the dust grain material and the imaginary part. The optical constants for amorphous SiO

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
(compare Fig. 3). Here, we find that this heating is
balanced by collisional cooling
.
Since both heating/cooling rates scale as
for large
Knudsen numbers and as
for small Knudsen numbers, a
constant value for
tunes in for both cases,
K for large *Kn*, and
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
for the calculation of
.

For larger particles (roughly at 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. . 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 is by far too small to reach the sublimation temperature , unless gas temperatures very close to are considered and, therefore, Eqs. (22) and (32) remain valid.

Copyright ESO 2003