How thick is the atmosphere? A derivation of the Boltzmann distribution

Note that while the temperature rockets up to 1000C in the upper atmosphere (thermosphere), it&#x27;s offset by the decrease in density: almost a thousand-fold between 100-150 km.<p>High temperature is not the same as hot. Those particles have a high temperature which means they&#x27;re moving around a lot (crude estimate from conservation of energy: v = √kT&#x2F;m). When you touch a &quot;hot&quot; 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&#x27;t sufficient heat transfer to burn you.<p>EDIT: There are other interesting considerations. Rocket controllers use precise atmospheric models to know when to throttle down&#x2F;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 &quot;Go at throttle up&quot; to let the pilot know that it&#x27;s safe to punch the engines. <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=Em-Krwbn25A" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=Em-Krwbn25A</a>

Also fun: &quot;The Kármán line... represents the boundary between Earth&#x27;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.&quot;<p><a href="https:&#x2F;&#x2F;www.wikipedia.org&#x2F;wiki&#x2F;Kármán_line" rel="nofollow">https:&#x2F;&#x2F;www.wikipedia.org&#x2F;wiki&#x2F;Kármán_line</a>

I&#x27;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.

Statistical mechanics is one of the most underrated Physics disciplines, having been largely replaced with the simpler thermodynamics in many undergrad courses.<p>If there was one Physical formula for humanity to leave behind, it should be Boltzmann&#x27;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&#x2F;nanoscale physics.

you had me at lagrange multipliers

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.

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