Please can someone tell me if my thinking here is right...
I've got a planet with an atmospheric pressure at 6km of 0.5 P0 and at 8km of 0.4 P0 (P0 = pressure at the surface).
I want to work out the scale height of the atmosphere.
Given scale height = λ
and for height above surface = z
I could rearrange to show the pressure at the surface as:
I could then use the relative pressure, assume P(0)=1 (as it will cancel out shortly) and height from each of the know quantities and set them equal to each other like this:
0.4/e(-8000/λ) = 0.5/e(-6000/λ)
A little mutliplication....
0.4 e(-6000/λ) = 0.5 e(-8000/λ)
Take the Log of both sides....
(-6000/λ) log 0.4 = (-8000/λ) log 0.5
But know I'm left with the λ cancelling out if I multiply both sides by λ. I'm sure I've gone wrong here somewhere. Probably something very simple. Can anyone advise? Have I made a simple mistake in my working or have I gone completely off the reservation and need to start again? I just need to end up with λ = xxx metres.