Shape, shear and flexion II - Quantifying the flexion formalism for extended sources with the ray-bundle method

Flexion-based weak gravitational lensing analysis is proving to be a useful adjunct to traditional shear-based techniques. As flexion arises from gradients across an image, analytic and numerical techniques are required to investigate flexion predictions for extended image/source pairs. Using the Schwarzschild lens model, we demonstrate that the ray-bundle method for gravitational lensing can be used to accurately recover second flexion, and is consistent with recovery of zero first flexion. Using lens plane to source plane bundle propagation, we find that second flexion can be recovered with an error no worse than 1% for bundle radii smaller than {\Delta}{\theta} = 0.01 {\theta}_E and lens plane impact pararameters greater than {\theta}_E + {\Delta}{\theta}, where {\theta}_E is the angular Einstein radius. Using source plane to lens plane bundle propagation, we demonstrate the existence of a preferred flexion zone. For images at radii closer to the lens than the inner boundary of this zone, indicative of the true strong lensing regime, the flexion formalism should be used with caution (errors greater than 5% for extended image/source pairs). We also define a shear zone boundary, beyond which image shapes are essentially indistinguishable from ellipses (1% error in ellipticity). While suggestive that a traditional weak lensing analysis is satisfactory beyond this boundary, a potentially detectable non-zero flexion signal remains.