A combination of faulty design of the solid rocket booster segment joints and deliberate progressive relaxation of launch safety procedures by NASA officials and the weather (the temperature that morning was the coldest of any shuttle launch ever).

The design error was to use rubber O-rings to attempt to seal a joint that expands significantly during normal operation, rubber O-rings should only be used where they can be kept constantly in compression. Also at the temperature on the morning of the launch the rubber became hard, losing all elasticity and thus had no ability to expand when the joints flexed.

NASA had documented cases of charred O-rings from hot gasses in the solid rocket boosters escaping past the O-rings and out the joints, yet they ignored this instead of trying to address it and fix the problem.

On the day of the launch of the Challenger, as there was the teacher Christa Mcauliffe was in the crew it was a "prestige flight" and very important and NASA officials deliberately ignored several safety warnings that would have required a launch delay!

Note: Unlike what most people think there was no explosion involved in the Challenger disaster, the vehicle was torn apart by flight stresses that exceeded design limits as the solid rocket booster crushed into the top of the external tank while the main engines gimbaled to their limits to try to keep the vehicle on course in response to uneven thrust from the two solid rocket boosters. The photos that appear to show a "fireball" engulfing the vehicle is just a condensing water vapor cloud (produced by burning of hydrogen and oxygen escaping from the damaged external tank) illuminated by sunlight.

The events of the launch leading to the disaster were:

main engine ignition and gimbal test
solid rocket booster ignition
the joint on one solid rocket booster burned through and the flame damaged the lower mount point for that solid rocket booster, weakening the mount point
ash and clinker plugged up the burned through joint, temporarily stopping the flame
liftoff
under the flight stressed of "main engine throttle up" the ash and clinker plugging the burned through joint failed and a strong flame escaped, causing a reduction in pressure in that solid rocket booster and a corresponding drop in its thrust
the resulting uneven thrust from the two solid rocket booster caused the vehicle to deviate from the required flight trajectory
the computer detected this course deviation and attempted to correct for it by gimballing the main engines and solid rocket booster nozzles, however they quickly hit the mechanical gimbal limits and could go no further while the vehicle continued to deviate from the required flight trajectory
the stresses from the maximum gimbaled engines broke the damaged lower mount point on that solid rocket booster
the solid rocket booster, now no longer constrained by the lower mount point pivoted around the upper mount point causing it to hit the top of the external tank and breach it
at this point the vehicle was now doomed with one solid rocket booster at a large off angle orientation and ripping into the external tank, the other solid rocket booster torquing the vehicle in one direction while the main engines were unable to correct this as their gimbals were at mechanical limits
now hydrogen and oxygen escaping from the damaged tank ignited producing water vapor, which as it condensed to form a cloud which obscured the vehicle from view, sunlight illuminated this cloud produced the false impression of an explosion "fireball"
the engine thrust torques mentioned above then rapidly broke up the shuttle and external tank
the crew capsule separated from the broken up shuttle and followed a ballistic free fall trajectory until impact with the water of the Atlantic where the enormous g-forces of impact killed the crew and crushed the crew capsule

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