The chain of events leading to the destruction of Ariane 5's maiden voyage, flight 501, began well before the launch. When the launch vehicle is sitting on its launch table, on-board computers use inertial measurements to make sure the vehicle knows its exact location on Earth. The rotation of the Earth and the sway of the vehicle in the wind must be taken into account so that the launcher can precisely place its payload in orbit. Instruments take measurements of the accelerations the vehicle is experiencing, and the Inertial Reference System (SRI) computer takes this information does a few operations on it, and then passes it to the on-board computer, which uses it to compute the position of the vehicle. Once the vehicle lifts off, it is no longer necessary to compute where the vehicle is on Earth, but the computers continue to analyse the data for about forty seconds after liftoff. This only occurs because it was required for Ariane 4, from which Ariane 5 inherited its tried and true SRI units. However, Ariane 5 operates on a different time line and flight path than did Ariane 4, and the software running the SRI units on Ariane 5 wasn't changed to take this into account. This was largely because the SRI units were believed to be the most reliable parts of Ariane 5 since they had already been extensively tested on Ariane 4.

After flight 501 lifted off, the vehicle experienced accelerations much greater than it did on the ground. Because the accelerations are much greater, the numbers used to represent accelerations in the SRI computers became very large. About 39 seconds after liftoff, the number related to horizontal acceleration became so large that when it was converted from a 64 bit floating point number to a 16 bit integer so that it could be passed to the on-board computer, an error occurred, which caused the SRI computers to shut down. Ariane 5 is equipped with two SRI units, so that one can take over if the other fails. In this case, both units failed in exactly the same way, so the data received by the on-board computer was not correct data, but instead a diagnostic pattern generated by the SRI computers after they had shutdown.

The on-board computer interpreted the diagnostic pattern as flight data, and that caused it to incorrectly command the engine nozzles on the solid rocket boosters and on the main engine to deflect as much as possible. For the rocket, this was like making a very sharp turn in a car that's moving very fast; the car will roll, or in Ariane 5's case, large aerodynamic forces caused the solid rocket boosters to separate from the main stage. At this point, Ariane 5 did what it was supposed to; it exploded.

An inquiry into the failure of flight 501 found the defect in the ADA code that performed the unprotected type cast, and the larger design defect that allowed that specific bit of code to run after Ariane 5 had lifted off. These defects were fixed, and haven't bothered Arianespace since.

References

Arianespace, Ariane 5 Failure - Full Report, 19 July 1996, http://www.esa.int/htdocs/tidc/Press/Press96/ariane5rep.html

Kunzig, Robert. "Europe's Dream." Discover, May 199?: ??.
(Try Discover.com, search the archives for Arianespace)

Node your homework!
One detail missing:

The payload of Ariane flight 501 was the Cluster satellites. This was a group of four satellites which would orbit in formation to measure solar winds in three dimensions. A successful launch would have meant setting a new record for the number of satellites launched simultaneously

After the failure of the launch, these satellites were destroyed. However, the European Space Agency succesfully launched Cluster II (essentially the same mission) in two seperate launches, on the 16th of July and the 9th of August 2000.

But this time, a Russian Soyuz rocket was used.

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