# 霍曼轉移軌道

## 計算

${\displaystyle E={\begin{matrix}{\frac {1}{2}}\end{matrix}}mv^{2}-{\frac {GMm}{r}}={\frac {-GMm}{2a}}}$

${\displaystyle v^{2}=\mu \left({\frac {2}{r}}-{\frac {1}{a}}\right)}$

• ${\displaystyle v\,\!}$為物體的速度
• ${\displaystyle \mu =GM\,\!}$為中央物體的標準重力參數
• ${\displaystyle r\,\!}$為物體至中央物體中心的距離
• ${\displaystyle a\,\!}$為物體軌道的半長軸

${\displaystyle \Delta v={\sqrt {\frac {\mu }{r_{1}}}}\left({\sqrt {\frac {2r_{2}}{r_{1}+r_{2}}}}-1\right)}$
${\displaystyle \Delta v^{\prime }={\sqrt {\frac {\mu }{r_{2}}}}\left(1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{2}}}}\,\!\right)}$

${\displaystyle r_{1}}$${\displaystyle r_{2}}$ 分別是原本圓軌道與目標圓軌道的半徑，其中大的（小的）對應到霍曼轉移軌道的遠拱點（近拱點）距離。

${\displaystyle t_{H}={\begin{matrix}{\frac {1}{2}}\end{matrix}}{\sqrt {\frac {4\pi ^{2}a_{H}^{3}}{\mu }}}=\pi {\sqrt {\frac {(r_{1}+r_{2})^{3}}{8\mu }}}}$

（即橢圓軌道週期的一半），其中${\displaystyle a_{H}\,\!}$是霍曼轉移軌道的半長軸。

## 參考文獻

• Walter Hohmann. Die Erreichbarkeit der Himmelskörper. Verlag Oldenbourg in München. 1925. ISBN 3-486-23106-5.
• Thornton, Stephen T.; Marion, Jerry B. Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole. 2003. ISBN 0-534-40896-6.
• Bate, R.R., Mueller, D.D., White, J.E.,. Fundamentals of Astrodynamics. Dover Publications, New York. 1971. ISBN 978-0486600611.
• Vallado, D. A. Fundamentals of Astrodynamics and Applications, 2nd Edition. Springer. 2001. ISBN 978-0792369035.
• Battin, R.H. An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics & Ast, Washington, DC. 1999. ISBN 978-1563473425.