# 氣壓公式

## 理想情況

{\displaystyle {\begin{aligned}&(P+dP)A+\rho Ag_{0}dz=PA\\&PA+AdP+\rho Ag_{0}dz=PA\\&AdP+\rho Ag_{0}dz=0\\&AdP=-\rho Ag_{0}dz\\&dP=-\rho g_{0}dz\\&{\frac {\partial P}{\partial z}}=-\rho g_{0}\end{aligned}}}

${\displaystyle {\frac {dP}{dz}}=-\rho g_{0}\cdots (1)}$

{\displaystyle {\begin{aligned}&PV=nR^{*}T\\&P\left({\frac {V}{n}}\right)=R^{*}T\\&P\left({\frac {\rho V}{n}}\right)=\rho R^{*}T\\&PM=\rho R^{*}T\\&\rho ={\frac {M}{R^{*}T}}P\cdots (2)\end{aligned}}}

${\displaystyle (1)}$${\displaystyle (2)}$ 合埋，變成：

{\displaystyle {\begin{aligned}&{\frac {dP}{dz}}=-\left({\frac {M}{R^{*}T}}P\right)g_{0}\\&{\frac {dP}{dz}}=-{\frac {g_{0}M}{R^{*}T}}P\end{aligned}}}

{\displaystyle {\begin{aligned}&{\frac {dP}{P}}=-{\frac {g_{0}M}{R^{*}T}}dz\\&\int _{P_{0}}^{P}{\frac {dP}{P}}=\int _{0}^{z}\left(-{\frac {g_{0}M}{R^{*}T}}\right)dz\\&[\ln |P|]_{P_{0}}^{P}=-{\frac {g_{0}M}{R^{*}T}}[z]_{0}^{z}\\&\ln P-\ln P_{0}=-{\frac {g_{0}M}{R^{*}T}}z\\&\ln {\frac {P}{P_{0}}}=-{\frac {g_{0}M}{R^{*}T}}z\\&{\frac {P}{P_{0}}}=e^{-{\frac {g_{0}M}{R^{*}T}}z}\\&P=P_{0}e^{-{\frac {g_{0}M}{R^{*}T}}z}\end{aligned}}}

## 實際情況

${\displaystyle PM=\rho R^{*}(T_{0}-L_{b}z)}$

{\displaystyle {\begin{aligned}&{\frac {dP}{dz}}=-{\frac {g_{0}M}{R^{*}(T_{0}-L_{b}z)}}P\\&{\frac {dP}{P}}=-{\frac {g_{0}M}{R^{*}(T_{0}-L_{b}z)}}dz\\&\int _{P_{0}}^{P}{\frac {dP}{P}}=\int _{0}^{z}\left[-{\frac {g_{0}M}{R^{*}(T_{0}-L_{b}z)}}\right]dz\\&[\ln |P|]_{P_{0}}^{P}={\frac {g_{0}M}{R^{*}}}\left[{\frac {1}{L_{b}}}\ln |T_{0}-L_{b}z|\right]_{0}^{z}\\&\ln {\frac {P}{P_{0}}}={\frac {g_{0}M}{R^{*}L_{b}}}\ln {\frac {T_{0}-L_{b}z}{T_{0}}}\\&{\frac {P}{P_{0}}}=\left({\frac {T_{0}-L_{b}z}{T_{0}}}\right)^{\frac {g_{0}M}{R^{*}L_{b}}}\\&P=P_{0}\left({\frac {T_{0}-L_{b}z}{T_{0}}}\right)^{\frac {g_{0}M}{R^{*}L_{b}}}\end{aligned}}}

## 根據參考高度而定嘅氣壓公式

${\displaystyle P=P_{0}e^{-{\frac {g_{0}M}{R^{*}T}}(z-z_{0})}}$ （假設恆溫大氣層）

${\displaystyle P=P_{0}\left[{\frac {T_{0}-L_{b}(z-z_{0})}{T_{0}}}\right]^{\frac {g_{0}M}{R^{*}L_{b}}}}$ （根據氣溫垂直遞減率）