Pressure in atmosphere

Introduction

ISO2533:1975

The case in Toposhere (<10 km).

• Lapse rate +6.5 °C/km
• Base temp 19.0 °C
• Base atmospheric pressure 108,900 Pa equals 1.075 atm
• Base atmospheric density 1.2985 kg/m3

International Standard Atmosphere

https://en.wikipedia.org/wiki/International_Standard_Atmosphere

Consider the function in GY91's BMP280-3.3, eg at https://startingelectronics.org/tutorials/arduino/modules/pressure-sensor/

Earth's atmosphere's changes in

• Pressure $P$
• Temperature $T$
• Density $\rho$
• Viscosity $\eta$ or $\mu$

Hydrostatic balance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dP}{dh} = -\rho g } The ideal gas law Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \rho R T }

Reference atmospheric model

How the ideal gas properties change (mainly) as a function of altitude (etc).

Static atmospheric model

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \frac{MP}{RT} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dP = -g \rho dh} (see above).

Isothermal-barotropic approximation and scale height

Temperature and molecular weight are constant: density and pressure are exponential functions of altitude.

The US standard atmosphere

More realistic temperature function, consisting of eight data points connected by straight lines, which is---of course---an approximation.

https://en.wikipedia.org/wiki/U.S._Standard_Atmosphere

Barometric formula

Models how the pressure of the air changes with altitude with linear temperature change.

Ideal gas law $P={\frac {\rho }{M}}RT$, where pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n, T, V} , thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=P(n,T,V)} and the hydrostatic assumption Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dP = - \rho g dz} are needed to derive this.

Assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {dp}{dh}=-\rho(h) g } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \rho \frac{R}{M}T} which gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \frac{pM}{RT}} and we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {dp}{dh}=- g \frac{pM}{RT} }

Actually, both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} are dependent on altitude $h$. Thus, we will assume linear dependency on temperature $T=T_{0}-Lh$, and we have

${\begin{aligned}{\frac {dp}{dh}}&=-g{\frac {pM}{R(T_{0}-Lh)}}\\{\frac {dp}{p}}&=-g{\frac {M}{R(T_{0}-Lh)}}dh\\\ln p&={\frac {gM}{RL}}\ln(T_{0}-hL)+C\\p&=p_{0}\left(T_{0}-hL\right)^{\frac {gM}{RL}}\end{aligned}}$

Simplified model from Weather.gov

https://www.weather.gov/media/epz/wxcalc/pressureAltitude.pdf

$h=44307.70\left(1-\left({\frac {p}{1013.25}}\right)^{0.190284}\right)$