# 扰动位

## 数学表达

${\displaystyle T=W-U}$

${\displaystyle \nabla T\equiv 0}$

### 球谐函数

${\displaystyle T(r,\theta ,\lambda )=\sum _{n=0}^{\infty }\left({R \over r}\right)^{n+1}T_{n}(\theta ,\lambda )}$

${\displaystyle T_{n}(\theta ,\lambda )={GM \over R}\sum _{m=0}^{n}\left(\Delta C_{nm}\cos {m\lambda }+\Delta S_{nm}\sin {m\lambda }\right)P_{nm}(\cos {\theta })}$

• ${\displaystyle GM}$  地心引力常数
• ${\displaystyle n}$ 为面谐函数的阶数
• ${\displaystyle R}$  为用以近似的地球半径
• ${\displaystyle P_{nm}}$ ${\displaystyle n\ }$ ${\displaystyle m\ }$ 次的缔合勒让德多项式
• ${\displaystyle \Delta C_{nm}}$ ${\displaystyle \Delta S_{nm}}$  是真实重力位与正常重力位所采用的球谐系数的差值

${\displaystyle T(r,\theta ,\lambda )=\sum _{n=2}^{\infty }\left({R \over r}\right)^{n+1}T_{n}(\theta ,\lambda )}$

## 径向导数

${\displaystyle {\partial T \over \partial r}=-{1 \over r}\sum _{n=2}^{\infty }(n+1)\left({R \over r}\right)^{n+1}T_{n}(\theta ,\lambda )}$
${\displaystyle {\partial ^{2}T \over \partial r^{2}}={1 \over r^{2}}\sum _{n=2}^{\infty }(n+1)(n+2)\left({R \over r}\right)^{n+1}T_{n}(\theta ,\lambda )}$

${\displaystyle r^{2}{\partial ^{2}T \over \partial r^{2}}=\sum _{n=2}^{\infty }\left({R \over r}\right)^{n+1}T'_{n}(\theta ,\lambda )}$

## 球面近似

${\displaystyle {\partial \over \partial n}={\partial \over \partial h}={\partial \over \partial r}}$

${\displaystyle n}$ ${\displaystyle h}$ ${\displaystyle r}$  分别表示重力矢量的方向、高程方向和地心方向。

### 大地水准面上的扰动位及其径向导数

${\displaystyle T=T(R,\theta ,\lambda )=\sum _{n=2}^{\infty }T_{n}(\theta ,\lambda )}$

## 与其他物理量的关系

### 重力扰动

${\displaystyle \delta {\vec {\text{g}}}={\vec {\text{g}}}_{P}-{\vec {\gamma }}_{P}}$

${\displaystyle \delta {\text{g}}=-({\partial W \over \partial n}-{\partial U \over \partial n'})\longrightarrow \delta {\text{g}}=-({\partial W \over \partial n}-{\partial U \over \partial n})=-{\partial T \over \partial n}\longrightarrow \delta {\text{g}}=-{\partial T \over \partial r}}$

${\displaystyle \delta {\text{g}}={1 \over r}\sum _{n=2}^{\infty }(n+1)\left({R \over r}\right)^{n+1}T_{n}(\theta ,\lambda )}$

### 重力异常

${\displaystyle \Delta {\vec {\text{g}}}={\vec {\text{g}}}_{P}-{\vec {\gamma }}_{Q}}$

${\displaystyle {\partial T \over \partial r}+{2 \over r}T+\Delta {\text{g}}=0\longrightarrow \Delta {\text{g}}=-{\partial T \over \partial r}-{2 \over r}T}$

${\displaystyle \Delta {\text{g}}={1 \over r}\sum _{n=2}^{\infty }(n-1)\left({R \over r}\right)^{n+1}T_{n}(\theta ,\lambda )}$

### 斯托克斯公式

${\displaystyle T={R \over 4\pi }\iint \limits _{\sigma }\Delta g\,S(\psi )\operatorname {d} \!\sigma }$

${\displaystyle S(\psi )={1 \over \sin(\psi /2)}-6\sin {\psi \over 2}+1-5\cos {\psi }-3\cos {\psi }\ln {(\sin {\psi \over 2}+\sin ^{2}{\psi \over 2})}}$

## 参考文献

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3. Torge, Wolfgang. Geodesy. Walter de Gruyter GmbH & Co KG. 2001. ISBN 978-3-11-017072-6 （英语）.
4. ^ 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.
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6. ^ 宁津生. 管泽霖 , 编. 地球形状及外部重力场. 测绘出版社. 1981.
7. ^ Survey, U. S. Coast and Geodetic; Lambert, Walter Davis; Darling, Frederic Warren. Tables for Determining the Form of the Geoid and Its Indirect Effect on Gravity. U.S. Government Printing Office. 1936 （英语）.