# What is the difference between these various Angular Resolution formulas?

I have recently learned about angular resolution and its dependence on the ratio λ/D. There seem to be three different equations for angular resolution that I have come across. One of these has the number 1.22 (radians), while the other contains the number 206,000 (arcseconds). A third equation states that angular resolution can be found just with the ratio λ/D. I have attached a picture of these three different equations - please pardon the bad merging!

Naturally, this has caused some confusion on what situations each of these equations apply to. Would one of these be more relevant to an interferometer — while another is for a single telescope? I'd like to share that these equations come from the textbooks "Foundations of Astrophysics" (Ryden & Peterson) and "Understanding Our Universe" (Palen, Kay & Blumenthal). If anyone has come across these equations, I'd be grateful for some advice on how to distinguish them (i.e. what situations or conditions they apply to).

$$\theta_\text{min}[\text{rad}] = 1.22 \frac{\lambda}{D}$$ $$\theta = 2.06 \times 10^5 \frac{\lambda}{D}~\text{arcseconds}$$

A telescope's resolution is determined by the ratio $$\lambda/D$$.

• Welcome to Stack Exchange! I've kept a link to the screen shot but written out the equation as a block quote using MathJax. Screen shots of text or equations are discouraged whenever the information can be written out explicitly. It helps in several ways; some people use screen text readers or other software, it can be easily copied and reused, and it can then be found using site searches.
– uhoh
Sep 20 '20 at 3:07
• @MikeG thanks for the fixer-upper. I attribute these glitches to "cosmic rays"
– uhoh
Sep 20 '20 at 14:28

$$2.06 \times 10^5$$ is the radian arcsecond conversion, equal to $$180 \times 60 \times 60 / \pi$$ .
The $$1.22$$ is a geometrical factor applicable to circular apertures, as outlined in Wikipedia.
The expression $$1.22 \frac\lambda D$$ (in radians) represents the distance between the center of an Airy disk and the first minimum of an Airy disk that results from a perfect lens with a perfectly circular aperture and a perfectly focused point source. That assumes a bit too much perfection for some, who also see using the center of the first minimum as a bit generous. In those people's minds, it is best to reduce the factor of 1.22 to (approximately) one, resulting in $$\frac\lambda D$$ as the angular resolution.
Regarding the factor of $$2.06\times10^5$$ arc seconds, that is one radian.