The Panchromatic Hubble Andromeda Treasury XI: The Spatially-Resolved Recent Star Formation History of M31

We measure the recent star formation history (SFH) across M31 using optical images taken with the \texit{Hubble Space Telescope} as part of the Panchromatic Hubble Andromeda Treasury (PHAT). We fit the color-magnitude diagrams in ~9000 regions that are ~100 pc $\times$ 100 pc in projected size, covering a 0.5 square degree area (~380 kpc$^2$, deprojected) in the NE quadrant of M31. We show that the SFHs vary significantly on these small spatial scales but that there are also coherent galaxy-wide fluctuations in the SFH back to ~500 Myr, most notably in M31's 10-kpc star-forming ring. We find that the 10-kpc ring is at least 400 Myr old, showing ongoing star formation over the past ~500 Myr. This indicates the presence of molecular gas in the ring over at least 2 dynamical times at this radius. We also find that the ring's position is constant throughout this time, and is stationary at the level of 1 km/s, although there is evidence for broadening of the ring due to diffusion of stars into the disk. Based on existing models of M31's ring features, the lack of evolution in the ring's position makes a purely collisional ring origin highly unlikely. We find that the global SFR has been fairly constant over the last ~500 Myr, though it does show a small increase at 50 Myr that is 1.3 times the average SFR over the past 100 Myr. During the last ~500 Myr, ~60% of all SF occurs in the 10-kpc ring. Finally, we find that in the past 100 Myr, the average SFR over the PHAT survey area is $0.28\pm0.03$ M$_\odot$ yr$^{-1}$ with an average deprojected intensity of $7.3 \times 10^{-4}$ M$_\odot$ yr$^{-1}$ kpc$^{-2}$, which yields a total SFR of ~0.7 M$_\odot$ yr$^{-1}$ when extrapolated to the entire area of M31's disk. This SFR is consistent with measurements from broadband estimates. [abridged]