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The rate of sublimation of a dust grain at temperature T is given (in g cm-2 s-1) by

Equation 7 (7)

where the index i runs over all equilibrium gas-phase species (e.g. C, C2, CO, etc. for graphite grains), pi is the partial pressure of species i, Si(T) ~ 1 is the sticking fraction for molecules colliding with the grain surface, and psi is the saturation vapor pressure for species i. The saturation vapor pressure is given by

Equation 8 (8)

where zetai is the product of the rotational, vibrational, and electronic partition functions for the gas-phase species i, and h0 is the heat of sublimation at zero temperature, on a per atom basis. Delta(T) is a complicated expression involving integrals of the specific heat of the solid (Reif 1965, p. 367). Graphite (h0 / k = 8.58 x 104 K to C, h0 / k = 9.82 x 104 K to C2, h0 / k = 9.45 x 104 K to C3, Kelley 1973) is the most refractory substance (with the exception of Tungsten!); silicate grains have h0 / k appeq 6.6 x 104 K. We have fitted the thermodynamic data and sublimation measurements of graphite in vacuum (Kelley 1973), and find that the saturation vapor pressure of the dominant gas-phase constituent is well fitted by

Equation 9 (9)

The gas pressure in an alpha accretion disk is

Equation 10 (10)

This pressure is comparable to the pressure ~ 10-2 dyn cm-2 in broad-line clouds. Since the solar abundance of carbon, C/H = 4 x 10-4, the partial pressure of carbon, were it all in the gas phase, would be

Equation 11 (11)

At radii ~ 1 pc, grains grow at an impressive rate: a / adot ~ 5n9 (a / 0.1 µm) yr; in fact the temperatures and densities are quite similar to those in red-giant winds where interstellar dust is believed to form. [But here the grain and gas temperatures need not be equal: the ratio of a grain's radiative cooling luminosity to the rate at which it exchanges energy with gas via collisions is L/H = 10n11-1 Tg,35 / TH,33/2, where the grain temperature is 103 Tg,3 K and the gas temperature and density are 103 TH,3 and 1011 n11 cm-3, respectively]. However, comparing with equation (7) and equation (9), we see that when the grain temperatures Tg exceed 2000 K, graphite grains will certainly begin to sublimate rather than grow. For Tg > 2100 K, the timescale for sublimation of a 1 µm graphite grain becomes shorter than 103 yr, the timescale on which in could be replenished by inflow. From figure 1, we see that this temperature is reached at ~ 0.3 pc in our fiducial quasar.

When the dust sublimates, the gas loses its primary opacity and coolant. As the temperature rises above ~ 3000 K, most common molecules are destroyed, and the opacity drops precipitously by several orders of magnitude (Alexander et al. 1983). The gas in the interior of the disk is then unable to remain in thermal equilibrium at temperatures 2000 ltapprox T ltapprox 7000 K, and must inevitably heat. Above ~ 104 K, the opacity rises abruptly to near its former level as hydrogen is ionized, providing the gas with a new thermal equilibrium state. Unless there is no warp (or down-scattering of radiation onto the disk from electrons or a jet), the transition disk at r < 0.3 pc is thus constrained by the heating from the central source to be optically thin, with T ~ 104 K, until r ltapprox 0.02 pc, when the incident flux can be carried by optically thick thermal emission, and the temperature will begin to rise above 104 K in the accretion disk. The absence of thermalized emission from material with temperatures 2000 ltapprox T ltapprox 7000 K provides a natural explanation for the minimum in nuLnu at nu = 1014.5 Hz (lambda = 1 µm) observed in almost all quasars (Neugebauer et al. 1989). [The reader may mentally add an accretion disk spectrum to the right of figure 2]. Since in this interpretation the frequency is a universal constant, determined (up to very slowly varying logarithms) by the heats of sublimation and dissociation of dust and molecules, and by the ionization of hydrogen, the minimum in the reradiated nuLnu will always be present, and observable unless filled in by starlight or a non-thermal contribution to the spectrum.

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