# 大气质量 (天文学)

## 計算大氣質量

### 天頂角和高度

${\displaystyle h=90^{\circ }-z\,.}$

### 平面-平行大氣層

${\displaystyle X=\sec \,z\,.}$

### 插值公式

${\displaystyle X=\sec \,z_{\mathrm {t} }\,\left[1-0.0012\,(\sec ^{2}z_{\mathrm {t} }-1)\right]\,,}$

Hardie (1962)引入多項式${\displaystyle \sec \,z-1}$

${\displaystyle X=\sec \,z\,-\,0.0018167\,(\sec \,z\,-\,1)\,-\,0.002875\,(\sec \,z\,-\,1)^{2}\,-\,0.0008083\,(\sec \,z\,-\,1)^{3}\,}$

${\displaystyle X=\left(\cos \,z+0.025e^{-11\cos \,z}\right)^{-1}\,,}$

${\displaystyle X={\frac {1}{\cos \,z+0.50572\,(6.07995^{\circ }+90^{\circ }-z)^{-1.6364}}}\,,}$

Young (1994)發展的：

${\displaystyle X={\frac {1.002432\,\cos ^{2}z_{\mathrm {t} }+0.148386\,\cos \,z_{\mathrm {t} }+0.0096467}{\cos ^{3}z_{\mathrm {t} }+0.149864\,\cos ^{2}z_{\mathrm {t} }+0.0102963\,\cos \,z_{\mathrm {t} }+0.000303978}}\,}$

Pickering (2002)發展的：

${\displaystyle X={\frac {1}{\sin(h+{244}/(165+47h^{1.1}))}}\,,}$

### 大氣模型

#### 非折射徑向對稱大氣

${\displaystyle s={\sqrt {R_{\mathrm {E} }^{2}\cos ^{2}z+2R_{\mathrm {E} }y_{\mathrm {atm} }+y_{\mathrm {atm} }^{2}}}-R_{\mathrm {E} }\cos \,z\,}$

${\displaystyle s={\sqrt {\left(R_{\mathrm {E} }+y_{\mathrm {atm} }\right)^{2}-R_{\mathrm {E} }^{2}\sin ^{2}z}}-R_{\mathrm {E} }\cos \,z\,}$

#### 均質的大氣層

${\displaystyle X={\frac {s}{y_{\mathrm {atm} }}}={\frac {R_{\mathrm {E} }}{y_{\mathrm {atm} }}}{\sqrt {\cos ^{2}z+2{\frac {y_{\mathrm {atm} }}{R_{\mathrm {E} }}}+\left({\frac {y_{\mathrm {atm} }}{R_{\mathrm {E} }}}\right)^{2}}}-{\frac {R_{\mathrm {E} }}{y_{\mathrm {atm} }}}\cos \,z\,.}$

${\displaystyle y_{\mathrm {atm} }={\frac {kT_{0}}{mg}}\,,}$

${\displaystyle T_{0}}$  = 288.15 K，${\displaystyle m}$  = 28.9644×1.6605×10−27 kg，並且${\displaystyle g}$  = 9.80665 m/s2，得到${\displaystyle y_{\mathrm {atm} }}$  ≈ 8435 m。取地球的半徑值為 6371 km，在地平的海平面大氣質量為：

${\displaystyle X_{\mathrm {horiz} }={\sqrt {1+2{\frac {R_{\mathrm {E} }}{y_{\mathrm {atm} }}}}}\approx 38.87\,.}$

${\displaystyle {\frac {R_{\mathrm {E} }}{y_{\mathrm {atm} }}}={\frac {X^{2}-1}{2\left(1-X\cos z\right)}}\,;}$

## 注释

2. ^ 在非常高的天頂角，大氣質量強烈的依賴當地的大氣條件，包括溫度、壓力，特別是地面附近的溫度梯度。此外，低空的消光也受到氣溶膠濃度及其垂直分布的強烈影響。許多作者告誡說，精確計算地平線附近的大氣質量幾乎是不可能的。
3. ^ Kasten和Young的公式起初是以海拔高度為依據${\displaystyle \gamma }$  是指：
${\displaystyle X={\frac {1}{\sin \,\gamma +0.50572\,(\gamma +6.07995^{\circ })^{-1.6364}}}\;;}$
在此文中，給出了天頂角，以與其它的公式一致
4. ^ Pickering (2002)使用Garfinkel (1967)最為準確度的參考。
5. ^ Although acknowledging that an isothermal or polytropic atmosphere would have been more realistic, Janiczek and DeYoung (1987) used the homogeneous spherical model in calculating illumination from the Sun and Moon, with the implication that the slightly reduced accuracy was more than offset by the considerable reduction in computational overhead.

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