Bias properties of logarithmic and asinh magnitudes at very low flux levels

The observation of flux sources near the limit of detection requires a care ful evaluation of possible biases in magnitude determination. Both the trad itional logarithmic magnitudes and the recently proposed inverse hyperbolic sine (asinh) magnitudes are considered. Formulae are derived for three dif ferent biasing mechanisms: the statistical spread of the observed flux valu es arising from e.g. measurement error; the dependence of these errors on t he true flux; and the dependence of the observing probability on the true f lux. As an example of the results, it is noted that biases at large signal- to-noise ratios R, at which the two types of magnitude are similar, are of the order of -(p + 1)/R-2, where the exponent p parametrizes a power-law de pendence of the probability of observation on the true flux.