which may be re-expressed as

Ip, = Ip0 exp(—'rp(zo, zi)/P) = Ip0Tp(p,zo,z,) (6.34)

where rp (zo, z,) is the optical thickness at wavenumber p (or optical depth if measured downwards from the top of the atmosphere) between z0 and z, for a vertical path and Tp(p,z0,z,) is the transmission from z0 to z, at angle d.

Now the thin slab at altitude z of thickness dz will also emit thermal radiation and from Kirchoff's law the emissivity of the layer is equal to its absorptivity. Hence, the spectral radiance emitted by this layer at angle d is equal to dIp = kp(z)n(z)Bp(z) dz/p, (6.35)

where Bp (z) is the Planck function at wavenumber p and altitude z, where the temperature is T(z). This spectral radiance is itself attenuated by overlying layers before reaching the top of the atmosphere and thus the contribution to Ip from this slab is kpn(z)Bp(z) dz dip =-exp

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