# 正常重力

## 分布情况

${\displaystyle \gamma _{e}}$  ${\displaystyle 9.780\,326\,7715\,{\text{m}}\cdot {\text{s}}^{-2}}$  椭球赤道处的正常重力值 [6]:117
${\displaystyle \gamma _{p}}$  ${\displaystyle 9.832\,186\,3685\,{\text{m}}\cdot {\text{s}}^{-2}}$  椭球极点处的正常重力值 [6]:117
${\displaystyle \gamma _{45}}$  ${\displaystyle 9.806\,199\,203\,{\text{m}}\cdot {\text{s}}^{-2}}$  椭球45°纬线处的正常重力值 [7]
${\displaystyle {\bar {\gamma }}}$  ${\displaystyle 9.797\,644\,656\,{\text{m}}\cdot {\text{s}}^{-2}}$  整个椭球面上的平均正常重力值 [7]

## 数学表达

${\displaystyle {\boldsymbol {\gamma }}=\nabla U}$

• ${\displaystyle \gamma _{u}={1 \over w}{\partial U \over \partial u}}$
• ${\displaystyle \gamma _{\beta }={1 \over w{\sqrt {u^{2}+E^{2}}}}{\partial U \over \partial \beta }}$
• ${\displaystyle \gamma _{\lambda }={1 \over {\sqrt {u^{2}+E^{2}}}\cos \beta }{\partial U \over \partial \lambda }=0}$

## 计算公式

• 椭球的半短轴 ${\displaystyle b=a(1-f)}$
• 椭球的第一偏心率 ${\displaystyle e={\sqrt {a^{2}-b^{2}}}/a=2f-f^{2}}$
• 椭球的第二偏心率 ${\displaystyle e={\sqrt {a^{2}-b^{2}}}/b}$

### 克莱罗定理

${\displaystyle f+f^{*}={5 \over 2}{\omega ^{2}b \over \gamma _{e}}(1+{9 \over 35}{e'}^{2})}$

• ${\displaystyle f^{*}={\gamma _{p}-\gamma _{e} \over \gamma _{e}}}$
• ${\displaystyle \gamma _{e}={GM \over a^{2}}\left(1+m+{3 \over 7}{e'}^{2}m\right)}$
• ${\displaystyle \gamma _{p}={GM \over ab}\left(1-{3 \over 2}m-{3 \over 14}{e'}^{2}m\right)}$

### 正常重力公式

#### 对称形式

${\displaystyle \gamma ={a\gamma _{p}\sin ^{2}\beta +b\gamma _{e}\cos ^{2}\beta \over {\sqrt {a^{2}\sin ^{2}\beta +b^{2}\cos ^{2}\beta }}}}$

${\displaystyle \tan \beta ={b \over a}\tan \varphi }$

${\displaystyle \gamma ={a\gamma _{e}\cos ^{2}\varphi +b\gamma _{p}\sin ^{2}\varphi \over {\sqrt {a^{2}\cos ^{2}\varphi +b^{2}\sin ^{2}\varphi }}}}$

#### 截断形式

${\displaystyle \gamma =\gamma _{e}(1+f_{2}\sin ^{2}\varphi +f_{4}\sin ^{4}\varphi )}$

• ${\displaystyle f_{2}=-f+{5 \over 2}m+{1 \over 2}f^{2}-{26 \over 7}fm+{15 \over 4}m^{2}}$
• ${\displaystyle f_{4}=-{1 \over 2}f^{2}+{5 \over 2}fm}$

${\displaystyle \gamma =\gamma _{e}(1+f^{*}\sin ^{2}\varphi -{1 \over 4}f_{4}\sin ^{4}2\varphi )}$

#### 闭合形式

${\displaystyle \gamma =\gamma _{e}{1+k\sin ^{2}\varphi \over {\sqrt {1-e^{2}sin^{2}\varphi }}}}$

${\displaystyle k={b\gamma _{p}-a\gamma _{e} \over a\gamma _{e}}}$

#### 数值形式

1979年 ${\displaystyle \gamma =9.780\,327\,(1+0.005\,3024\sin ^{2}\varphi -0.000\,0058\sin ^{4}2\varphi )\,{\text{m}}\cdot {\text{s}}^{-2}}$  ${\displaystyle 0.1\,{\text{mgal}}}$  [7]

### 向上延拓公式

${\displaystyle \gamma _{h}=\gamma +{\partial \gamma \over \partial h}h+{1 \over 2}{\partial ^{2}\gamma \over \partial h^{2}}h^{2}+\cdots }$

${\displaystyle {\partial \gamma \over \partial h}=-2\gamma J-2\omega ^{2}=-{2\gamma \over a}\left(1+f+m-2f\sin ^{2}\varphi \right)}$

${\displaystyle {\partial ^{2}\gamma \over \partial h^{2}}={6GM \over a^{4}}={6\gamma \over a^{2}}}$

${\displaystyle \gamma _{h}=\gamma \left[1-{2 \over a}\left(1+f+m-2f\sin ^{2}\varphi \right)h+{3 \over a^{2}}h^{2}\right]}$

${\displaystyle \gamma _{h}=\gamma -0.3086h+0.72\times 10^{-7}h^{2}}$

## 注释

1. ^ 其中 ${\displaystyle u}$  表示椭球的半短轴${\displaystyle \beta }$  表示归化纬度${\displaystyle \lambda }$  表示经度

## 参考文献

1. ^ GeographicLib: Normal gravity. geographiclib.sourceforge.io. [2020-04-13].
2. Somigliana, Carlo. Teoria generale del campo gravitazionale dell'ellissoide di rotazione. Libr. Editr. Politecnica. 1929 （意大利语）.
3. ^ 宁津生. 管泽霖 , 编. 地球形状及外部重力场. 测绘出版社. 1981: 154–293.
4. ^ Jekeli, C. Potential Theory and Static Gravity Field of the Earth. Treatise on Geophysics. Elsevier. 2007: 11–42. ISBN 978-0-444-52748-6. doi:10.1016/b978-044452748-6.00054-7 （英语）.
5. 孔祥元; 郭际明; 刘宗泉. 大地测量学基础. 武汉大学出版社. 2001. ISBN 978-7-30-707562-7.
6. Torge, Wolfgang. Geodesy. Walter de Gruyter GmbH & Co KG. 2001. ISBN 978-3-11-017072-6 （英语）.
7. H. Moritz. Geodetic Reference System 1980 (pdf) （英语）.
8. San Francisco W. H. Freeman and Company. Heiskanen Moritz 1967 Physical Geodesy. San Francisco: W. H. Freeman and Company. 1967 （英语）.
9. ^ Alexis Claude, Clairaut. Théorie de la figure de la terre : tirée des principes de l'hydrostatique (PDF). 1743.
10. Department of Defense World Geodetic System 1984 ― Its Definition and Relationships with Local Geodetic Systems (pdf). [2020-04-14] （英语）.
11. ^ Vanícek, P.; Krakiwsky, E. J. Geodesy: The Concepts. Elsevier. 2015-06-03. ISBN 978-1-4832-9079-9 （英语）.