Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2022 / 4 / c / Solution

Ignoring refraction, the line of sight is perpendicular to the radius at its tangent point. For a surface tangent, Pythagorean theorem gives the satellite-to-tangent distanceFor tangent altitude $z_t$, replace $R^2$ by $(R+z_t{)}^2$; lower atmospheric scale heights change the estimate little. The required angular resolution isUsing the Rayleigh criterion for a circular telescope aperture, $\theta =1.22\lambda /d$. The longest wavelength imposes the strongest condition, soThe corresponding requirement at $7\mu \mathrm{m}$ is $3.3$ mm. This is a diffraction limit, without allowance for scan sampling, aberrations or instrumental field of view.