extract low-resolution spectra from diffraction spikes

In imaging from a telescope with a secondary on a spider (for example, in HST imaging), bright stars show diffraction spikes. More generally, the outer parts of the point-spread function are related to the Fourier Transform of the small-scale features in the entrance aperture. The scale at which this Fourier Transform imprints on the focal plane is linearly related to wavelength (just as the angular size of the diffraction-limited PSF goes as wavelength over aperture).

This means that the diffraction spikes coming from stars contain low-resolution spectra of those stars! That is, you ought to be able to extract spectral information from the spikes. It won't be good, but it should permit measurements of colors or temperatures or SED slopes with even single-band imaging, and aid in star–quasar classification. Indeed, in HST press-release images, you can see that the diffraction spikes are little "rainbows" (see below).

The project is to take wide-band imaging from HST, in fields where stars have been measured either in multiple bands or else spectroscopically, and show that some of the scientific results could have been extracted from the single, wide band directly using the diffraction features.

measure the speed of light with Kepler

Okay this idea is dumb but I would love to see it done: As Kepler goes around the Sun (no, not the Earth, the Sun), it is sometimes flying towards its field and sometimes away. This leads to classical stellar aberration (discovered by Bradley in the 1700s; Bradley was a genious, IMHO), which leads to a beaming effect, in which the field-of-view (or plate scale) changes with the projection of the velocity vector onto the field-center pointing vector. A measurement of this would only take a day or two of hard work, and would provide a measure of the speed of light in units of the velocity of Kepler in its orbit.

Kepler as an insanely expensive thermometer!

The Kepler spacecraft is taking incredibly precise photometric data on tens of thousands of stars for the purpose of detecting exoplanets. For many reasons, the lightcurves it returns are sensitive to the temperature of the spacecraft: The focus and astrometric map (camera calibration) of the camera changes with temperature, and the detector noise properties might be evolving too. This wouldn't be a problem (it's a space mission) but the spacecraft changes its sun angle abruptly to perform high-gain data downlink about once per month, and the temperature recovery profile depends on the orientation of the spacecraft post-downlink. Instead, there are sub-percent-level traces of the temperature history imprinted on every lightcurve. Each lightcurve responds to temperature differently, but each is sensitive.

Of course the spacecraft keeps housekeeping data with temperature information, but it hasn't been extremely useful for calibration purposes. Why not? The onboard temperature sensors are low in signal-to-noise or dynamic range, whereas the lightcurves are good (sometimes) at the part-in-hundred-thousand level. That is, there is far more temperature information in the lightcurves than in the direct temperature data! Here's the project:

Treat the housekeeping data about temperature as providing noisy labels on the lightcurve data. Find the properties of each lightcurve that best predicts those labels. Combine information from many lightcurves to produce an extremely high signal-to-noise and precise temperature history for the spacecraft. Bonus points for constraining not just the temperature history but a thermal model too.