Line Caustic Revisted: Which distance is $δ$ in $J^{-1}\propto \sqrt{δ^{-1}}$?

Details Line Caustic Revisted: Which distance is $δ$ in $J^{-1}\propto \sqrt{δ^{-1}}$? Line Caustic Revisted: Which distance is $δ$ in $J^{-1}\propto \sqrt{δ^{-1}}$?

2002-05-05

arxiv.org/abs/astro-ph/0205067v1

Astronomy and Astrophysics

Astrophysics

31pages including 4figures

Scientific paper

The line caustic behavior has been discussed since Chang and Refsdal (1979) mentioned inverse-square-root-of-the-distance dependence of the amplification of the images near the critical curve in a study of a single point mass under the influence of a constant shear due to a larger mass. A quarter century later, Gaudi and Petters (2001) interprets that the distance is {\it a vertical distance to the caustic}. It is an erroneous misinterpretation. We rehash Rhie and Bennett (1999) where the caustic behavior of the binary lenses was derived to study the feasibility of limb darkening measurements in caustic crossing microlensing events. ~({\it 1}) $J = \pm \sqrt{4\delta\omega_{2-} J_-}$ where ~$\delta\omega\parallel\bar\partial J$, and $\delta\omega_{2-}$ and $J_-$ are $E_-$-components of $\delta\omega$ (the source position shift from the caustic curve) and $2\bar\partial J$ (the gradient of the Jacobian determinant) respectively; ~({\it 2}) The critical eigenvector $\pm E_-$ is normal to the caustic curve and easily determined from the analytic function $\kappa$-field; ~({\it 3}) Near a cusp ($J_- = 0$) is of a behavior of the third order, and the direction of $\bar\partial J$ with respect to the caustic curve changes rapidly because a cusp is an accumulation point; ~({\it 4}) On a planetary caustic, $|\partial J|\sim \sqrt{1/\epsilon_{pl}}$ is large and power expansion does not necessarily converge over the size of the lensed star. In practice, direct numerical summation is inevitable. We also note that a lens equation with constant shear is intrinsically incomplete and requires supplementary physical assumptions and interpretations in order to be a viable model for a lensing system.

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