
How thick is the atmosphere? A derivation of the Boltzmann distribution - simulate
https://gravityandlevity.wordpress.com/2018/07/14/how-thick-is-the-atmosphere-a-derivation-of-the-boltmzann-distribution/amp/
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joshvm
Note that while the temperature rockets up to 1000C in the upper atmosphere
(thermosphere), it's offset by the decrease in density: almost a thousand-fold
between 100-150 km.

High temperature is not the same as hot. Those particles have a high
temperature which means they're moving around a lot (crude estimate from
conservation of energy: v = √kT/m). When you touch a "hot" thing, the heat you
feel is all those particles bumping into your body. So in the upper atmosphere
those particles might be very excited, but the low density means there isn't
sufficient heat transfer to burn you.

EDIT: There are other interesting considerations. Rocket controllers use
precise atmospheric models to know when to throttle down/up during a launch.
There is a point called maximum dynamic pressure (max Q) where the stress on
the airframe is maximised due to the speed of the vehicle and local air
pressure. On Shuttle launches you hear the controllers call out "Go at
throttle up" to let the pilot know that it's safe to punch the engines.
[https://www.youtube.com/watch?v=Em-
Krwbn25A](https://www.youtube.com/watch?v=Em-Krwbn25A)

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AstralStorm
Unless you're moving faster than possible for your particles to return to
equilibrium. (either via conduction or via radiation) After all, heat transfer
is a large scale statistical phenomenon.

So, if you're moving fast, despite the low density you're hitting a lot of
(anisotropic) fast moving particles. This is a difference between static
friction and dynamic collisions.

This is why space ships still need heat shields bigger than expected by just
friction alone.

Edit: yes it is related to maximum dynamic pressure, you could call it maximum
heat differential.

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RhysU
Also fun: "The Kármán line... represents the boundary between Earth's
atmosphere and outer space.... Theodore von Kármán... was the first person to
calculate that the atmosphere around this altitude becomes too thin to support
aeronautical flight, since a vehicle at this altitude would have to travel
faster than orbital velocity to derive sufficient aerodynamic lift to support
itself."

[https://www.wikipedia.org/wiki/Kármán_line](https://www.wikipedia.org/wiki/Kármán_line)

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jessriedel
Is this actually a well-defined altitude, or does it depend on the properties
of the vehicle? (Certainly, the minimum speed to maintain level flight
normally does.) The Karman line is near but distinct from the boundary between
the thermosphere and the exosphere, which is well-defined: it's where atoms
become so thin that their mean free path becomes so long they stop bouncing
off each other and just follow parabolic trajectories under gravity.

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dotancohen
I'd like to upvote this twice. I love articles that explain scientific
principles in a way that I could explain them to my 11 year old daughter. This
article is a perfect example, balancing the equations that I want to see with
the text that she needs to grasp the concept without understanding the
equations. These random never-thought-about-the-physics-behind-that articles
are terrific.

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asafira
You have a gifted daughter.

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guitarbill
Statistical mechanics is one of the most underrated Physics disciplines,
having been largely replaced with the simpler thermodynamics in many undergrad
courses.

If there was one Physical formula for humanity to leave behind, it should be
Boltzmann's entropy formula. Conceptually linking microstates to macrostates
could lead to a glut of other discoveries, including thermodynamics (think
Industrial Revolution), solid-state (semiconductor stuff), and
molecular/nanoscale physics.

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jules
Aren't statistical physics and thermodynamics complementary? Thermodynamics is
about deriving results given an equation of state and statistical physics is a
method for finding the equation of state.

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Mvandenbergh
Yes, thermodynamics is rooted in empirical results and the use of
thermodynamics in industrial age engineering pre-dates the statistical
mechanics explanations of why the laws of thermodynamics are true.

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wolfi1
you had me at lagrange multipliers

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selimthegrim
A similar derivation is given in the first edition of Kittel’s Thermal
Physics, as well as in the Quantum Mechanics book by Griffiths IIRC.

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asafira
Does Kittel define Shannon entropy?

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selimthegrim
I'd have to grab the first edition copy from the library but I think not, he
does use Lagrange multipliers though.

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kevinwang
Should K.E. be labeled as vertical kinetic energy in the first set of
equations instead of vertical potential energy?

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asafira
Yes, you are right. It's a typo.

