[Physics] Optimal telescope size

astronomydiffractionopticstelescopes

Consider a diffraction-limited telescope with unobstructed aperture $D$. Such a scope is capable of yielding an angular resolution $\alpha$ that scales as $\lambda/D$, with $\lambda$ denoting the wavelength of the light. However, in reality such a telescope will need to look through a turbulent atmosphere, the refractive index variations of which will cause wavefront errors with a spatial correlation length (Fried parameter) $r_0$. As a result, no stable diffraction pattern but rather a speckle pattern will form. I am interested in the angular resolution corresponding to the average size of this speckle pattern.

What is this angular resolution $\alpha(D/\lambda,r_0/\lambda)$?

More specifically: how does $\alpha$ scale for $D\gg r_0$, and for fixed $r_0$: is there an optimal (highest resolution) aperture $D$? Or more down-to-earth: If I built a 200 inch Hale telescope in my backyard*, would it beat a 10 inch amateur telescope in terms of optical resolution?

*You may assume my backyard to have a typical seeing characterized by a Fried parameter $r_0$ of about 4 inch.

If your telescope is diffraction limited, then your angular resolution is $1.22 \lambda / D$. A reasonable limit to the size of your telescope would be to set the seeing disk FWHM equal to your angular resolution, which for the best case (blue light, $\lambda \approx 400 ~\text{nm}$) would give you a diameter of 4 inches (10 cm), and in the worst case (red light, $\lambda \approx 700 ~\text{nm}$) a diameter of 7 inches (17.6 cm). So, if you build a backyard telescope without any adaptive optics, you only need to build a 7 inch diameter telescope to achieve maximum angular resolution!