You probably expected a joke, but you've got to remember that I've spent all my brain capacity trying to get us to Lajaland. Anyways, now that I have your attention...
Did you ever wonder why the fuck we spent so much time on previous post trying to figure out what gases we had in the atmosphere of Lajaland?
It was to create a model of the atmosphere!!
And why? To have a model on how the atmosphere will affect our landing, example regarding air resistance. This will help us plan landing.
Warning: This post will include a lot of heavy duty math. To keep you awake along the way I have included memes, drawings, colours and TLDR (too lazy didn't read) at the bottom of each math section.
NASA: After completing the atmospheric model you are expected to have found the density profile and the temperature profile
Before we begin looking at this model, there are some assumptions to be made:
- The atmosphere is uniform. Equal distribution of each type of gas in the atmosphere.
- The atmosphere is spherically symmetric.
- The atmosphere is in hydrostatic equilibrium (will be explained later)
- Approximate the atmosphere as an ideal gas (will also be explained why later)
- The atmosphere is 50/50 adiabatic and isothermal (there will be a recap later)
The way this post is structured is me presenting to you a bunch of math, piece by piece. You will probably be confused along the way. Then at the end I will combine everything together and everything will make sense.You will say "AHA!" and I say "Tada! An atmosphere model."
Let's begin!
Average molecular weight (\(\mu\))
We failed to analyse flux to figure out what Lajaland's atmosphere consisted of. So this is an assumption of what it may consist of, based on what gases atmosphere usually has.
TLDR: We assume our atmosphere has H2O, CO2 and O2 (water, carbon dioxide and oxygen). The atmosphere of Lajaland has therefore an average molecular weight of\(\mu \approx 31.094\) in units of hydrogen mass.
Hydrostatic equilibrium (and pressure)
TLDR: Only when the equation for hydrostatic equilibrium (which I just deduced) is fulfilled do we have an atmosphere because all forces on the atmosphere equalise each other. Essentially, this is also an expression for pressure in the atmosphere.
Ideal gas law (density)
The reason we assumed the atmosphere was an ideal gas was because we already have the ideal gas law. Which makes finding an expression for pressure easier.
TLDR: I just found an expression for pressure as a function of distance r from planet surface. But don't I already have an expression for pressure?? Well if you take that equation and do what norwegians call "move and switch" (algebraic manipulation in english) we end up with an expression for density (\(\rho\), rho).
Temperature profile (T)
TLDR: I found the temperature profile as a function of r with regard to a 50/50 adiabatic and isothermal atmosphere.
Now comes what should have been the spectacular ending where everyone said AHA!, right?
Unfortunately the sun has risen and there are birds chirping outside of my window. I should sleep.
BUT I WILL COME WITH A CONCLUSION.
The idea of introducing these equations for pressure, density and temperature was because if you combine them together with... yes, more math... we should end up with a model of the atmosphere. Either as a numerical differential equation (which can be solved with Euler Cromer) or analytical integral.
NASA: Did you find the density profile and the temperature profile?
Rebecca: Only the equations :) but I can tell you what I expect the temperature profile and density profile to look like.
Temperature: From the equation we derived, the temperature decreases in the adiabatic part up until T0/2, where we enter the isothermal part. Fom here it remains constant
Density: Expected to decrease with increasing altitude because of less air pressing you down from above. That's why your ears get clogged in airplanes (I think)
Okay good night.
