I been trying to find the way I can calculate the telescope lens diameter needed to see a star by the distance of the star or the planet from the earth. Is there any mathematical relation we usually use to build our telescopes lens?
Example : Proxima Centauri b is about 4.24 light years away from us. We need a telescope that will capture its atmosphere. How big would the lens of that telescope needed will be?
The relation for angular resolution (angle $\theta$), the wavelength $\lambda$ and the telescope opening $D$ is simple, given by the diffraction formula for the Airy disk:
$$ \sin \theta \approx \theta = 1.22 \frac{\lambda}{D}$$
The angle $\theta$ can be calculated from the triangle from observer to object-of-interest (distance $d$ and its extend $r$: $\theta \approx r/d$.
For a distance of $d = 4.24 \mathrm{LY} = 4.01\cdot 10^{16}$m, you get for 1000km-size features an angle of $\theta = \frac{1,000,000\mathrm{m}}{4.01\cdot 10^{16}\mathrm{m}} = 2.5\cdot 10^{-11}$, and thus $$ D = 1.22\cdot \frac{\lambda}{\theta} = 1.22\frac{500\cdot 10^{-9}}{2.5\cdot 10^{-11}} = 25000\mathrm{m} = 25\mathrm{km}$$
Yet you don't need to resolve the atmosphere or planet at all in order to detect and analyse it: you can do spectroscopy and compare the host star's spectrum to the planet's spectrum. The difference will be the influence of the planet's atmosphere. This is already being done.
