## GNU Astronomy Utilities

#### 7.4.3.2 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.