## GNU Astronomy Utilities

#### 7.4.3.1 Magnitude measurement error of each detection

The raw error in measuring the magnitude is only meaningful when the object’s magnitude is brighter than the upper-limit magnitude (see below). As discussed in Brightness, Flux, Magnitude and Surface brightness, the magnitude ($$M$$) of an object with brightness $$B$$ and zero point magnitude $$z$$ can be written as:

$$M=-2.5\log_{10}(B)+z$$

Calculating the derivative with respect to $$B$$, we get:

$${dM\over dB} = {-2.5\over {B\times ln(10)}}$$

From the Tailor series ($$\Delta{M}=dM/dB\times\Delta{B}$$), we can write:

$$\Delta{M} = \left|{-2.5\over ln(10)}\right|\times{\Delta{B}\over{B}}$$

But, $$\Delta{B}/B$$ is just the inverse of the Signal-to-noise ratio ($$S/N$$), so we can write the error in magnitude in terms of the signal-to-noise ratio:

$$\Delta{M} = {2.5\over{S/N\times ln(10)}}$$

MakeCatalog uses this relation to estimate the magnitude errors. The signal-to-noise ratio is calculated in different ways for clumps and objects (see Akhlaghi and Ichikawa [2015]), but this single equation can be used to estimate the measured magnitude error afterwards for any type of target.