Xi = 1/v/2no-i is the mean free path; and ai is the collision cross-section. Hence, the coefficient may be written as Di = ci/3^/2nai or more generally in the semi-empirical form
nn where bi is the binary collision parameter expressed in terms of the coefficients At and Si, which are fitted to experimental data.
For an atmosphere in hydrostatic equilibrium (Section 4.1.1), the pressure distribution p(z) is given by p = p0e-z/H, where the scale height is to a first approximation H = RT/mg (assuming negligible variation of temperature with height) and m is the mean molecular weight. Similarly, it is straightforward to show that the number density n—(z) = p(z)/kT has a similar distribution given by n- = n0e -z/H (3.15)
where H* is the number density scale height, given by
Substituting for niE(z) into Equation (3.12) we obtain
where we have also equated the flux to the product of the number density times the mean vertical flux velocity wi. A mean estimate for the diffusion time is given by the expression Ti ~ H*/wt or, since wt is of the order of D/H, Ti ~ H2/Di.
Defining the volume mixing ratio, or mole fraction, of the i th element as fi = ni/na, and differentiating we obtain
fi dz ni dz na dz
Substituting this into Equation (3.17) we get h = ^ - DinadfZ (3.20)
where the limiting flux ftL is given by
Hence, exospheric loss is in fact moderated by the rate at which molecules may diffuse up to the exobase. This consideration is of particular importance for estimating the rate of escape of H and H2 from the atmospheres of Titan and Triton. Note that in Equations (3.20) and (3.21) the eddy diffusion coefficient should also be added to the molecular diffusion coefficient to give total "diffusion". Eddy diffusion is discussed in Chapter 4.
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