Measuring galaxy cluster total masses and the amount of dark matter substructure within galaxy cluster haloes is a fundamental probe of the $Lambda$CDM model of structure formation, as well as the interactions between baryonic and non-baryonic matter. Weak gravitational lensing is a technique for measuring the total mass distribution which is agnostic to the nature of the gravitating matter. We present a novel method for measuring weak gravitational lensing flexion, the gradient of the lensing shear field, to inform mass measurements on small scales. While previously published methods for measuring flexion focus on measuring derived properties of the lensed images, such as shapelet coefficients or surface brightness moments, this method fits a mass-sheet-invariant, parametrized Analytic Image Model (AIM) to the each galaxy image. The simple parametric model traces the distortion of lensed image isophotes and constrains small scale mass variations. We test the AIM method using simulated data images with realistic noise and a variety of input image properties, and show that it successfully reproduces the input lensing fields. We also apply the AIM method for flexion measurement to Hubble Space Telescope observations of Abell 1689, and detect mass structure in that cluster using only flexion measured with the AIM method.
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