The frequency integration extends over the far-ultraviolet regime above 6 eV, while the factor 4n comes from geometric considerations. For an isotropic specific intensity, Jv = Iv, the flux impinging on a planar surface element would be / Jv¡d,Q = 2n J0 Jvidi = nJv. However, we picture standard grains as being (roughly) spherical in shape. The additional factor of 4 in equation (7.17) is simply the ratio of the surface area of a sphere to the cross-sectional area which that sphere presents to a plane-parallel flux. The quantity ad is the cross section in the latter sense.
We saw in Chapter 2 how £d = ndad/nH is determined, through a combination of empirical and theoretical steps, to be 1.5 x 10-21 cm2. The quantity 4n /FUV Jv dv is estimated from observations to be 1.6 x 10-3 erg cm-2 s-1, a figure traditionally known as the "Habing flux." Using a representative ePE of 0.01 in equation (7.17), we find that rPE has the value 2 x 10-11 (nH/103 cm-3) eVcm-3 s-1. This estimate, however, ignores the influence of very small grains, including PAHs. For these, the ejected electrons can more easily reach the surface, i. e., the efficiency factor is higher. In addition, the number density of such grains is relatively high. Recall that the abundance of 10 A grains exceeds that of 0.1 pm particles by (10-7/10-5)-3 5 = 107. Despite the small cross section of each PAH, their cumulative effect is thus appreciable. By integrating over a realistic grain size distribution, we arrive at a final heating rate of rPE = 3 x 10-11 ( ) eVcm-3«-1 . (7.18)
V103 cm-3/
This figure assumes that each grain is electrically neutral, a condition that is violated in sufficiently strong ultraviolet radiation fields. We will consider the resulting alteration to rPE when we discuss the photodissociation regions near massive stars in Chapter 8.
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