Ummm ... remember back in differential equations that boundary values mattered? That's why the classic Boyce and DiPrima textbook in the field is called Differential Equations and Boundary Value Problems.
It turns out that climate scientists are using an 80 year-old solution technique for their differential equations which assumes a boundary value consistent with our atmosphere being infinitely thick.
That's right - no satellites because they'd all burn up from the friction. You'd think that would bug them. Apparently not.
Why use this approximation at all? Well, it works for some applications:
In stellar atmospheres ... this could be a reasonable assumption, and ... has great practical value in astrophysical applications.
Last I checked though, the prefix "astro" means out there, not down here. Oh ... and we don't live on the surface of a star.
Down here, that approximation leads to a little bit of mathematical goofiness that comes out of all the global warming models: the air temperature can be persistently higher than the ground temperature.
That's a problem because in reality they've got to balance out in the long-run where they touch.
In the models, they don't.
This is really convenient, because it allows the models to generate bigger increases in atmospheric temperatures than can be reconciled with ground temperatures.
Yes, you are reading this correctly: global warming models suggest that you won't burn your hands if you hold them over a boiling kettle because your skin temperature doesn't have to reach the same temperature as the steam. Like I said ... goofy.
But, in our world, satellites orbit freely because our atmosphere does end. What happens if you resolve the problem with a boundary consistent with a finite atmosphere?
Well, for one, without the infinite atmosphere (to mop up the excesses) the air temperature reaches the ground temperature at the surface.
Even better, this is caused by an additional feedback mechanism that isn't present in the simpler model (so this theory is general in that it contains the standard theory as a special case).
The effect of that feedback mechanism is to put a strong upper bound on the temperature increase associated with an increase in atmospheric carbon dioxide. And that bound isn't too far off where our temperatures are now.
The behavior of the model in response to shocks is interesting too: an increase in carbon dioxide causes temperatures to spike up quickly, followed by a gradual decline. Sound familiar? Especially given the news from last year that NASA was nudging up the temperature data from the past decade to make it looks like the temperature spike of the 80s and 90s was still going on.
It gets better. The research was published in a peer-reviewed journal by a 30 year veteran of NASA who resigned because he felt his views were being suppressed. Go read it for yourself - it's accessible for someone with some technical skills.