# 冷澤-提爾苓進動

## 數學推導

${\displaystyle {\boldsymbol {B}}={\frac {3}{5}}R^{2}q{\Big (}{\boldsymbol {\omega }}\cdot {\boldsymbol {r}}{\frac {\boldsymbol {r}}{r^{5}}}-{\frac {1}{3}}{\frac {\boldsymbol {\omega }}{r^{3}}}{\Big )}}$

${\displaystyle {\boldsymbol {\omega }}=-4\int {\frac {\rho {\boldsymbol {u}}\,dV}{r}}}$

${\displaystyle {\boldsymbol {B}}={\frac {12}{5}}R^{2}q{\Big (}{\boldsymbol {\omega }}\cdot {\boldsymbol {r}}{\frac {\boldsymbol {r}}{r^{5}}}-{\frac {1}{3}}{\frac {\boldsymbol {\omega }}{r^{3}}}{\Big )}}$

${\displaystyle {\boldsymbol {B}}=-\left({\frac {1}{3}}{\frac {\boldsymbol {\omega }}{r^{3}}}\cos \theta \right)}$

${\displaystyle {\boldsymbol {B}}=-{\frac {4}{5}}{\frac {{\boldsymbol {\omega }}mR^{2}}{r^{3}}}\cos \theta }$

${\displaystyle \Omega _{\text{LT}}=-{\frac {2}{5}}{\frac {Gm\omega }{c^{2}R}}\cos \theta }$

${\displaystyle \Omega _{\text{LT}}=-2.2\cdot 10^{-4}}$角秒/日。

${\displaystyle \Omega _{\text{rel}}={\frac {3\pi Gm}{c^{2}r}}}$

## 參考資料

1. ^ Pfister, H. On the history of the so-called Lense–Thirring effect. General Relativity and Gravitation. November 2007, 39 (11): 1735–1748. Bibcode:2007GReGr..39.1735P. doi:10.1007/s10714-007-0521-4.