# A Calculating a Telescope's Limiting Magnitude

1. Mar 12, 2016

### Ishtar_UK

Hello Everyone

I'm new to this community so I hope that I have posted this question to the correct forum. If not, please advise on the best forum for my question.

Recently, I have been trying to find a reliable formula to calculate a specific telescope's limiting magnitude while factoring magnification, the telescopes transmission coefficient and the observers dilated pupil size. It's clear that higher magnifications darken the sky and so increase contrast (which has the effect of increasing the limiting magnitude). Also, the observers pupil size has a substantial effect too and thus must be taken into account.

The standard limiting magnitude calculation can be expressed as:

LM = 2.5 * LOG10( (Aperture / Pupil_Size)2 ) + NELM

But obviously this does not take into account the darkening effects of magnification. After reading the Wikipedia page on Limiting Magnitude I came up with the following equation.

LM = (NELM - 2) + 2.5 LOG10( ((Aperture / Pupil_Size)2) * power)

However, the equation above is exaggerating answer. In fact, it is about two magnitudes too high. I am also not sure why the Wikipedia article was suggested subtracting 2 from the NELM, but without it, the answer is even further out.

I did some further digging around and managed to come up with the following equation which does give a reliable limiting magnitude while factoring the telescopes transmission but does not include the observers pupil size.

Stage One

Darkening_Factor =
28.57 - 2.814 * NELM + 0.369 * NELM2 + 5 * LOG10( power / (aperture * √ transmission_coefficient) )

Stage Two

LM =
-22.81 + 1.792 * Darkening_Factor - 0.02949 * Darkening_Factor2 + 2.5 * LOG10( Aperture2 ) * transmission_coefficient)

As stated above, this seems to work well but does not include the observers pupil size. I have played around with the equation by doing the following but sadly this does not seem to work. Also, I am not sure what all those constants mean in stage one and two.

Stage Two - Modified: Does NOT Work

LM =
-22.81 + 1.792 * Darkening_Factor - 0.02949 * Darkening_Factor2 + 2.5 * LOG10( (Aperture / pupil_size)2) * transmission_coefficient)

I would be grateful if somebody could point me in the right direction. I believe I am nearly there, that is, aside from reliably including the observers pupil size.

Any help would be gratefully received.

Amanda

2. Mar 14, 2016

### Chronos

Last edited by a moderator: May 7, 2017
3. Mar 14, 2016

### Ishtar_UK

Thank you Chronos

I am aware of the online calculator, I've been using it to test my calculations. However, for various reasons, want to make these calculations myself. I will take a look at the Limiting Magnitude article; it looks just what I am looking for.

Many Thanks
Amanda

4. Mar 14, 2016