  Cometary activity in main-belt asteroids probes the ice content of these
objects and provides clues to the history of volatiles in the inner solar
system. We search the Palomar Transient Factory (PTF) survey to derive upper
limits on the population size of active main-belt comets (MBCs). From data
collected March 2009 through July 2012, we extracted 2 million observations of
220 thousand known main-belt objects (40% of the known population, down to 1-km
diameter) and discovered 626 new objects in multi-night linked detections. We
formally quantify the extendedness of a small-body observation, account for
systematic variation in this metric (e.g., due to on-sky motion) and evaluate
this method's robustness in identifying cometary activity using observations of
115 comets, including two known candidate MBCs and six newly-discovered
non-main-belt comets (two of which were originally designated as asteroids by
other surveys). We demonstrate a 66% detection efficiency with respect to the
extendedness distribution of the 115 sampled comets, and a 100% detection
efficiency with respect to extendedness levels greater than or equal to those
we observed in the known candidate MBCs P/2010 R2 (La Sagra) and P/2006 VW139.
Using a log-constant prior, we infer 95% confidence upper limits of 33 and 22
active MBCs (per million main-belt asteroids down to 1-km diameter), for
detection efficiencies of 66% and 100%, respectively. In a follow-up to this
morphological search, we will perform a photometric (disk-integrated
brightening) search for MBCs.
