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Kepler's immense photometric precision to date was maintained throughsatellite stability and precise pointing. In this white paper, we argue thatimage modeling--fitting the Kepler-downlinked raw pixel data--can vastlyimprove the precision of Kepler in pointing-degraded two-wheel mode. We arguethat a non-trivial modeling effort may permit continuance of photometry at10-ppm-level precision. We demonstrate some baby steps towards precise modelsin both data-driven (flexible) and physics-driven (interpretably parameterized)modes. We demonstrate that the expected drift or jitter in positions in thetwo-weel era will help with constraining calibration parameters. In particular,we show that we can infer the device flat-field at higher than pixelresolution; that is, we can infer pixel-to-pixel variations in intra-pixelsensitivity. These results are relevant to almost any scientific goal for therepurposed mission; image modeling ought to be a part of any two-wheelrepurpose for the satellite. We make other recommendations for Kepleroperations, but fundamentally advocate that the project stick with its coremission of finding and characterizing Earth analogs. [abridged]

Maximizing Kepler science return per telemetered pixel: Detailed models of the focal plane in the two-wheel era