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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 $\lambda/D$.

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  • $\begingroup$ 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. $\endgroup$
    – uhoh
    Sep 20, 2020 at 3:07
  • $\begingroup$ @MikeG thanks for the fixer-upper. I attribute these glitches to "cosmic rays" $\endgroup$
    – uhoh
    Sep 20, 2020 at 14:28

2 Answers 2

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$2.06 \times 10^5$ 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.

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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.

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