How do scientists determine the thickness of the atmosphere?
The thickness of the atmosphere is dependent on the exact definition you choose for what the atmosphere is.
The atmosphere is the envelope of gas surrounding the Earth. As you get higher and higher you enter into layers of the atmosphere where individual atoms can be so far apart that they might travel several kilometers before they collide, and the temperature (heat of the atoms/molecules) is over 2000 degrees but anyone at that point would freeze to death since the particles would not interact with you. This is the thermosphere and marks the upper limit of what scientists normally consider to be the atmosphere.
Higher than this is the exosphere which is a layer that is transitional between extremely thin atmosphere and outer space.
Therefore, the limit could be 1000km as this is roughly where the base of the Exosphere is and it is the point where particles collide so rarely that they no longer behave as a gas.
The limit could also be considered 10000km as this is the point where Earth's corona is visible from outer space when viewed in teh UV spectrum.
A final limit could be around 175,000km or roughly half the distance to the moon. This is the point that the force of the solar wind would be higher than the force of Earth's gravity, ie. the Earth would no longer hold any atmospheric particles past this point.