Scale per pixel

One of the important information in astrophotography is a scale (arcsec) per pixel. Usually, the lowest angular diameter per pixel you can achieve is amateur astrophotography is 0.5 arcsec/pixel, while 1-2 arcsec/pixel is considered comfortable to work with.

How to calculate it? On the internet you can find the following formula for it:

\[ scale\ [ arcsec / pixel ] = \frac{ pixel\ size\ [\mu m] \cdot 206.265}{ focal\ length\ [mm] } \]

If you are like me, you would like to know what is 206.265 constant and why result of dividing \( \mu m \) by \( mm \) is \( arcsec/pixel \).

So let's start from the beginning and try to figure out how to calculate scale per pixel; after all, it is a simple geometry.

As we focus in infinity (see hyperfocal distance), we can assume that the image plane is exactly at focal length \( f \) from the lens. Of course, \( \alpha \) is on both sides, so we end up with the following formula to calculate it:

\[ \alpha = 2\cdot\arctan \frac{\overline{AB}/2}{f} \]

For small angles, and this one is small for sure, we can assume that \( \arctan{\alpha} = \alpha \).

\[ \alpha = 2\cdot\frac{\overline{AB}}{2\cdot f} \]

\[ \alpha = \frac{\overline{AB}}{f} \]

We are almost there, but where is mythical 206.265. This result is in radians, and we prefer results in arcseconds.

\[ 1\ radian = \frac{360 \degree}{2\pi} = \frac{360\cdot 3600\ arcsec }{2\pi} \approx 206\,264\ arcsec\]

\[ \alpha = \frac{\overline{AB}}{f} [radians] = \frac{\overline{AB}}{f} \cdot 206\,264\ [arcsec] \]

Let's put initial variables names:

\[ scale\ [ arcsec / pixel ] = \frac{ pixel\ size\ [mm] \cdot 206\,265}{ focal\ length\ [mm] } \]

Please note that in the above formula we measure pixel size in millimeters, pixel size is usually in micrometers, so we need to divide the formula by 1,000.

\[ scale\ [ arcsec / pixel ] = \frac{ pixel\ size\ [\mu m] \cdot 206.265}{ focal\ length\ [mm] } \]

As a reward for reading so far, here is an online calculator. It doesn't simplify the equation and uses \( \arctan \) so can be used for bigger angles (like to calculate FOV of the whole matrix). Enjoy!