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The temperature difference is caused by the pressure difference. Airplanes have always caused pressure differentials. Jet engines just cause more pressure than wings and propellers do.

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5 points
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It has next to nothing to do with pressure, let alone temperature drop due to expansion. There are 2 things:

  1. When each one quantity of cold and warm air mix, the temperature of the mixture is almost exactly the midpoint (average), as the heat capacity is almost a constant.
  2. Vapor pressure of the water is a function of temperature and scales FAR more than linear.

So now when the hot, humid (burned hydrocarbon) air of the exhaust mixes with cold air the temperature drops a bit, but the vapor pressure drops massively. When conditions are right, the vapor pressure is now below the amount of vapor pressure that is actually present -> condensation.

vapor pressure over temperature data, note how it changes more than 2 orders of magnitude over only 100 K.

Just found this from NASA.

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