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The Kármán line is the international definition for the edge of space. It is located 100 km (62 miles) above the Earth's surface, around where the Aurora Borealis forms. The only country that takes its own alternative definition is the United States, which defines the edge of space as 50 miles (80 km) above the Earth's surface. The international astronautics and aeronautics body which supports the Kármán line is the Fédération Aéronautique Internationale (FAI).
The Kármán line idea originates with Theodore von Kármán, a Hungarian-American physicist who calculated that above around 100 km in the atmosphere, the air becomes so thin that a craft must travel at greater than orbital speed to stay aloft. This is too thin for aeronautic purposes, and as such, activity above 100km is demarcated as astronautic rather than aeronautic.
Another factor is placing the Kármán line where it is is its nearness to the mesosphere-thermosphere boundary, which lies at around 85 km 53 miles). The thermosphere is so named because it has a higher temperature than the layers below it, due to the ionizing ultraviolet radiation from the Sun. Although local temperatures in some parts of the thermosphere can top the temperature of fire, you could stand in the open in the region with a spacesuit, because the atoms are so far apart that their energy is not a real problem. Above, the thermosphere, in the ionosphere, the ionizing radiation is so prevalent that the entire layer is charged, which permits radio waves to be bounced off of it.
Above 100 km is when orbiting the Earth becomes possible, though it is easier and safer in the neighborhood of 220 km (137 miles), where we can find the International Space Station. Orbits below the Kármán line quickly degrade, approaching or slamming into the surface of the Earth. Above 100 km, a sustained orbit is possible, though orbital speed must be maintained to avoid degradation. When the Kármán line was established, numerous scientists made the relevant calculations to determine where the line was, and when the results agreed on 100 km, they eagerly agreed on it as an official demarcation. The fact that 100 km is an easy-to-remember number helped it for its future designation as the Kármán line.
Two questions. First of all, why is it so difficult to define exactly where earth atmosphere ends and space begins? There is obviously some dispute as the United States has adopted a different definition than the rest of the world. And that leads to the second question -- why does the American definition of where space begins differ from the one held by the rest of the world?