6. SUBLIMATION AND THE TEMPERATURE GAP

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

(7)

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

(8)

where _i 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. (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 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

(9)

The gas pressure in an accretion disk is

(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

(11)

At radii ~ 1 pc, grains grow at an impressive rate: a / ~ 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 T_g,3⁵ / T_H,3^3/2, where the grain temperature is 10³ T_g,3 K and the gas temperature and density are 10³ T_H,3 and 10¹¹ n₁₁ cm^-3, respectively]. However, comparing with equation (7) and equation (9), we see that when the grain temperatures T_g exceed 2000 K, graphite grains will certainly begin to sublimate rather than grow. For T_g > 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 T 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 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 T 7000 K provides a natural explanation for the minimum in L at = 10^14.5 Hz ( = 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 L will always be present, and observable unless filled in by starlight or a non-thermal contribution to the spectrum.