Formula for scale height

Integrating:

=> pressure falls off exponentially with height in atmosphere with uniform temperature.

has the dimensions of distance and is called the ‘scale height’.

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This is the expression for pressure in a non-ionized atmosphere. If the atmosphere is ionized, then the same number of ions and electrons must be present to maintain neutrality. If hydrogen is the only constituent of the gas, then each proton and electron act as independent particles of mass:

(m(p)+m(e-))/2 ~ m(p)/2.

Thus p = p_0 exp (- (m(p)gh) / 2kT )

The formula shows that pressure falls off exponentially with height in an atmosphere with a uniform temperature.