Education

Abstract

``We are on line of position 157-337. Will repeat this message on 6210 kilocycles. We are running north and south.” --Amelia Earhart on 2 July 1937 in her last transmission to Coast Guard cutter Itasca This was the last verified transmission from Amelia Earhart and her navigator Fred Noonan as they were crossing over the Pacific Ocean from Lae to Howland Island in an attempt to fly around the world along an equatorial route. Their disappearance, despite the most exhaustive maritime search in history, made them two of the most famous missing people in the world. Arguably, had their wreckage been rapidly discovered, the artifacts would have led to a clear understanding of what happened to them, and their loss would be much less notable in aviation history. Then and now, we accept failure as long as we learn from it. It never sits well with the public when the best technology, sound reasoning, and concerted effort fails to answer questions like, “Where are they?'' and “What happened to them?” Almost eighty years later, we are considerably more advanced. Long distance commercial airline travel is incredibly safe. Crew members can communicate with an air traffic control facility at any time. Using sophisticated satellite, inertial, radar, and VHF navigation equipment, airline crew and ground stations can determine an aircraft's precise position relatively easily. In fact, commercial aircraft have features which automatically communicate with ground stations at regular intervals. And yet in March of 2014, a state-of-the-art commercial jetliner carrying 239 souls disappeared over the Pacific, and once again, despite the best efforts of several nations, the crash site has yet to be discovered. Similar to the disappearance of Amelia Earhart almost 80 years ago, determining the most likely trajectory and final position is a complex mathematical problem. This Education section features a self-contained module by John Zweck that walks through the mathematics underlying the search for the crash site of MH370. Since MH370 flew for more than six hours after the last known communication between the pilot and air traffic control, the primary evidence in the search is the pings from the automated Aircraft Communications Addressing and Reporting System. Since these pings travel to a satellite and then to a ground station, analysis of ping travel time and Doppler shift provides some information about the aircraft position and speed. The article contains a very understandable derivation that maps the aircraft trajectory on a sphere and satellite positions to ping duration and Doppler shift. Each model for the aircraft's trajectory needs to be reconciled with actual data to determine model parameters, ultimately leading to a classic optimization problem. Using the ideas in this article together with official data from MH370, one can compute something very close to the priority search area used by the Australian Transportation Safety Board, the organization that is coordinating the underwater search. In the time since this article was received, wreckage from MH370 has washed ashore on Réunion Island and later other scattered locations, consistent with Lagrangian trajectories from the priority search area. The underwater search still continues and hopefully will be more successful than the search for Earhart's missing Lockheed Electra 10E, bringing closure to the families of the victims and to a public that still wonders what happened. This article is self-contained and appropriate for an advanced undergraduate course or a project. One interesting feature of the official inquiry into the MH370 disappearance is that the investigation team publicly released the data and their reasoning. This allowed many groups to explore the data and verify the official determination of the most likely trajectory. In addition to the scientific and public relations benefits to this approach, the release of the data had the unforeseen benefit of allowing exciting problems like these to quickly enter the classroom and inspire students with compelling problems.