A Modified Magnitude System that Produces Well-Behaved Magnitudes, Colors, and Errors Even for Low Signal-to-Noise Ratio Measurements

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Abstract
We describe a modification of the usual definition of astronomical magnitudes, replacing the usual logarithm with an inverse hyperbolic sine function; we call these modified magnitudes "asinh magnitudes." For objects detected at signal-to-noise ratios of greater than about 5, our modified definition is essentially identical to the traditional one; for fainter objects (including those with a formally negative flux), our definition is well behaved, tending to a definite value with finite errors as the flux goes to zero. This new definition is especially useful when considering the colors of faint objects, as the difference of two "asinh" magnitudes measures the usual flux ratio for bright objects, while avoiding the problems caused by dividing two very uncertain values for faint objects. The Sloan Digital Sky Survey data products will use this scheme to express all magnitudes in their catalogs.
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