The probability distribution functions of emission line flux measurements and their ratios

Abstract

Many physical parameters in astrophysics are derived using the ratios of two observed quantities. If the relative uncertainties on measurements are small enough, uncertainties can be propagated analytically using simplifying assumptions, but for large normally distributed uncertainties, the probability distribution of the ratio become skewed, with a modal value offset from that expected in Gaussian uncertainty propagation. Furthermore, the most likely value of a ratio A/B is not equal to the reciprocal of the most likely value of B/A. The effect is most pronounced when the uncertainty on the denominator is larger than that on the numerator. We show that this effect is seen in an analysis of 12,126 spectra from the Sloan Digital Sky Survey. The intrinsically fixed ratio of the [O III] lines at 4959 and 5007 ${\AA}$ is conventionally expressed as the ratio of the stronger line to the weaker line. Thus, the uncertainty on the denominator is larger, and non-Gaussian probability distributions result. By taking this effect into account, we derive an improved estimate of the intrinsic 5007/4959 ratio. We obtain a value of 3.012 $\pm$ 0.008, which is slightly but statistically significantly higher than the theoretical value of 2.98. We further investigate the suggestion that fluxes measured from emission lines at low signal to noise are strongly biased upwards. We were unable to detect this effect in the SDSS line flux measurements, and we could not reproduce the results of Rola and Pelat who first described this bias. We suggest that the magnitude of this effect may depend strongly on the specific fitting algorithm used.