The short answer is that Amelia Earhart and her companion on the fateful flight, Fred Noonan, vanished without a trace. If Earhart was flying the course she radioed to Itasca at the time of her disappearance, she was on a line which missed Howland Island by more than five miles. The chart used by Noonan misplaced the tiny Howland Island by that distance. In the visual conditions at the time the island would not have been seen. Though a remote possibility the Electra could reach Gardner Island exists, it is mathematically unlikely. The most rational explanation is the aircraft ran out of fuel and crashed into the Pacific, where it sank. No trace of wreckage or oil slick was ever found.
That explanation remains unsatisfactory for those who admire Earhart’s flying skills (which are exaggerated). The theories touched on here are but a few of many involving conspiracies among the high command of the American Pacific military forces, the War Department and subsequent Department of Defense, and the White House. Earhart’s final radio transmissions indicated she knew she was in trouble, in an unknown position, and running perilously low on fuel. There was no call of an emergency, no notification the airplane was about to ditch in the sea. Nor was there notice of an attempt to land on an unknown island or coral reef. She simply flew into history.
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