User:AKAF/Equations

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${\frac {dm}{m}}=-\left({\frac {d\left({\frac {V^{2}}{2}}\right)+gdr}{g_{0}I_{sp}V\left(1-{\frac {D+D_{e}}{F}}\right)}}\right)$

${\frac {dm}{m}}=-\left({\frac {d\left({\frac {V^{2}}{2}}\right)+gdr}{\eta _{0}h_{PR}\left(1-{\frac {D+D_{e}}{F}}\right)}}\right)$

$\eta _{0}={\frac {g_{0}r_{0}\left(1-{\frac {1}{2}}{\frac {r_{0}}{r}}\right)}{h_{PR}\left(1-{\frac {D+D_{e}}{F}}\right)\ln \left({\frac {1}{\Pi _{e}+{\frac {1}{\Gamma }}}}\right)}}$

$\eta _{0}={\frac {g_{0}r_{0}\left(1-{\frac {1}{2}}{\frac {r_{0}}{r}}\right)}{h_{PR}\left(1-{\frac {D+D_{e}}{F}}\right)\ln \left({\frac {1}{1-\Pi _{f}}}\right)}}$

$I_{sp}={\frac {\sqrt {g_{0}r_{0}}}{g_{0}\left(1-{\frac {D+D_{e}}{F}}\right)\ln \left({\frac {1}{1-\Pi _{f}}}\right)}}$

$\eta _{0}={\frac {g_{0}V_{0}}{h_{PR}}}\cdot I_{sp}={\frac {\mbox{Thrust Power}}{\mbox{Chemical energy rate}}}$

$\Pi _{f}=1-exp\left[-{\frac {\left({\frac {V_{initial}^{2}}{2}}-{\frac {V_{i}^{2}}{2}}\right)+\int {g}\,dr}{\eta _{0}h_{PR}\left(1-{\frac {D+D_{e}}{F}}\right)}}\right]$

$\Pi _{f}=1-exp\left[-{\frac {g_{0}r_{0}\left(1-{\frac {1}{2}}{\frac {r_{0}}{r}}\right)}{\eta _{0}h_{PR}\left(1-{\frac {D+D_{e}}{F}}\right)}}\right]$

$\Pi _{f}=1-e^{-BR}$

$B={\frac {g_{0}}{\eta _{0}h_{PR}\left(1-\phi _{e}\right){\frac {C_{L}}{C_{D}}}}}$

in the form of the Breguet range formula

$\Pi _{f}=1-exp\left[-{\frac {g_{0}R}{\eta _{0}h_{PR}\left(1-\phi _{e}\right){\frac {C_{L}}{C_{D}}}}}\right]$