Observing in 2-D Gets a Bit More Complicated

We are bounded in a nutshell of Infinite Space: Reading #1: Observing in 2-D Gets a Bit More Complicated

The primary source for information throughout this semester will be Astrophysics in a Nutshell by Dan Maoz, where he introduces the reader to the basics of Astrophysics and the new world we are now embarking on get to know. Among the first topics discussed, we have the explanations of how Astronomy primarily uses the Electromagnetic Spectrum to conduct its observations of the deep fields of space. We also come to realize how Astrophysics is one of the few disciplines that requires interaction of many fields of study in order to more fully comprehend the phenomenon we are observing, be it chemical, physical, optical, relativistic, geological, meteorological, or perhaps biological.

Within the optical, there are many factors that come into play when observing, and how each of these are accounted for and used to better images depends on knowledge of physics, atmospheric science, and a foundation of quantum mechanics where needed. As we saw in a previous exercise (), an essential part of optical science in Astronomy is the use of telescopes and how these resolve images. In this same exercise, we were able to assume the resolution of a set of receptors in order to achieve interferometry is based on the equation \[\theta = \frac{\lambda}{D},\] where D is the distance between the receptors, \(\lambda\) is the wavelength, and \(\theta\) is the angle of resolution.

However, this equation only describes the resolution in one dimension, and when attempting to resolve in two dimensions the equation becomes \[ \theta = 1.22 \frac{\lambda}{D} .\] This factor of 1.22 has its origins in the calculations of Sir George Airy in the nineteenth century, who found how the diffraction pattern found in observations of stars and other objects, one of many concentric rings, could be accounted for mathematically.  His descriptions are now known as the Airy Disk, accounting for the maxima and minima of an image and how these change the expression for angular resolution. (Carroll & Ostlie, 2007, 146-147).

This factor comes from an integration of a Fourier Transformation in 2D, which makes as much sense to me as it does to you. But, there is a simpler way to understand this factor. Accept that there are these series of differential equations called the Bessel functions, and one of these accurately describes these concentric rings in a series of solutions to the equation. The solutions are : \[ J_1 = 3.8317 , 7.0156, 13.3237…\] and these can be used to define \[ ka \sin \theta = J_1\] where k is the wave number, a is the radius of the separation of the receptors, and \(\theta \) is the angle of resolution. 

Using some substitution and small angle approximation, we get: \[ \theta = \frac{3.83}{ka}\] where k is defined as \[k = 2\pi / \lambda\] so \[ \theta = \frac{3.83 \lambda}{2\pi a} \] \[ \theta = 1.22 \frac{\lambda}{D}\] and we now understand where this 1.22 factor comes from.