On estimating the force-freeness based on observed magnetograms

It is a common practice in the solar physics community to test whether or not measured photospheric or chromospheric vector magnetograms are force-free, using the Maxwell stress as a measure. Some previous studies have suggested that magnetic fields of active regions in the solar chromosphere are close to be force-free whereas there is no consistency among previous studies on whether magnetic fields of active regions in the solar photosphere are force-free or not. Here we use three kinds of representative magnetic fields (analytical force-free solutions, modeled solar-like force-free fields and observed non-force-free fields) to discuss on how the measurement issues such as limited field of view, instrument sensitivity and measurement error could affect the estimation of force-freeness based on observed magnetograms. Unlike previous studies that focus on discussing the effect of limited field of view or instrument sensitivity, our calculation shows that just measurement error alone can significantly influence the results of force-freeness estimate, due to the fact that measurement errors in horizontal magnetic fields are usually ten times larger than that of the vertical fields. This property of measurement errors, interacting with the particular form of force-freeness estimate formula, would result in wrong judgments of the force-freeness: a truly force-free field may be mistakenly estimated as being non-force-free and a true non-force-free field may be estimated as being force-free. Our analysis calls for caution when interpreting the force-freeness estimates based on measured magnetograms, and also suggests that the true photospheric magnetic field may be further away from being force-free than they currently appear to be.