circular limit, spectroscopy

I spent part of the morning doing the linear-in-eccentricity limit of low eccentricity Kepler radial velocity problem with Price-Whelan and Jagannath. We had a slow start but eventually got expressions for the fourier expansion to m=2 for small-eccentricity systems. After that, Price-Whelan plotted them up and showed them to be as accurate as we expected (that is, residuals were order e²).

In the afternoon, Roweis and I got fired up about spectroscopy. Here are the first few words of a document I started writing up about our project:

In a modern spectrograph, light from fibers or slits is dispersed onto a two-dimensional CCD or CCD-like detector, with (usually) one direction on the detector corresponding to an angular displacement on the sky and another (usually) close-to-orthogonal direction corresponding to wavelength. There are also slitless spectrographs, where one direction is a mixture of angular position and wavelength. The problem of spectroscopic data reduction is the problem of extracting the one-dimensional spectra—astronomical source flux densities as a function of wavelength—from the two-dimensional images. There is a literature on "optimal extraction" of these one-dimensional spectra; the best extraction methods treat the two-dimensional image pixels as data to be fit by a model that consists of a one-dimensional spectrum laid out geometrically on the device and convolved with a two-dimensional point-spread function (which might be a function of the atmospheric seeing or the device properties or both).

Although the optimal extraction literature solves some important problems, the hardest part of spectroscopic data reduction lies not in the extraction step but in the step of measuring or learning the geometric and point-spread functions themselves. These functions have various names in the literature but they can be expressed with a single rectangular matrix A...