Once I realized this wasn't an aircraft, I knew it was most likely an orbiting object, and even more interesting was that this orbiting object had apparently flared twice in a single pass.
(Aside: A satellite flare is a highly specular reflection of the sun off of solar panels or other glossy surfaces on an orbiting spacecraft. The spacecraft must have a very special alignment to direct the rays of the sun to an observer on the dark side of the Earth, so seeing one flare, let alone two distinct flares in a row is quite rare.)
The aerospace engineer in me took over and I set about doing some back-of-the-envelope orbit determination to be sure I was looking at a satellite. I looked up the nearby stars on my trusty star chart and estimated the observed angular distance traveled during the 50 second flare interval. I then applied some trigonometry and the definition of mean motion, and I was able to ascertain that the body was indeed moving at the correct angular rate to be in low-Earth orbit. Just by observing the direction the flares pointed in, I knew it was a highly inclined orbit also. My coarse calculations alone didn't help me identify the specific object, but they at least confirmed that what I had captured could most likely be attributed to an orbiting body.
Exactly identifying the body by continuing in this manner seemed daunting . At best, my calculations would perhaps narrow the list of potential objects down to a few hundred. I knew estimating the other orbital elements based on my crude observations would be difficult to impossible. To make matters worse, the internal clock on my camera was significantly off (and had since been corrected), so the image timestamps were useless for finding any absolute timing information. I quickly realized this could turn into a rather laborious exercise of parsing and sorting Space Track data.
Fortunately, I remembered that my favorite planetarium software, Stellarium, was bundled with a plugin that seamlessly pulls two-line elements from Celestrak to display satellites passing overhead. I knew approximately when I was out shooting that night (based on when I sent some text messages as I was packing up), and sure enough, I was able to quickly find a spacecraft that exactly fit the observed trajectory: