פרמטרי התנועה לפי תנ"ז ואנרגיה

L=mvr

E=\frac{1}{2}mv^{2}-\frac{GMm}{r}=\frac{L^{2}}{2mr^{2}}-\frac{GMm}{r}

\frac{1}{r^{2}}-\frac{2GMm^{2}}{L^{2}}\frac{1}{r}-\frac{2mE}{L^{2}}=0

\frac{1}{r_{\pm}}=\frac{GMm^{2}}{L^{2}}\mp\sqrt{\left(\frac{GMm^{2}}{L^{2}}\right)^{2}+\frac{2mE}{L^{2}}}=\frac{GMm^{2}}{L^{2}}\left(1\mp\sqrt{1+\frac{2EL^{2}}{G^{2}M^{2}m^{3}}}\right)

\frac{1}{r_{\pm}}=\frac{1}{l}\left(1\mp\varepsilon\right)

l=\frac{L^{2}}{GMm^{2}}\qquad\varepsilon=\sqrt{1+\frac{2EL^{2}}{G^{2}M^{2}m^{3}}}

a=\frac{r_{max}+r_{min}}{2}=\frac{1}{2}\left(\frac{l}{1-\varepsilon}+\frac{l}{1+\varepsilon}\right)=\frac{l}{1-\varepsilon^{2}}=\frac{L^{2}}{GMm^{2}}\left(-\frac{G^{2}M^{2}m^{3}}{2EL^{2}}\right)=-\frac{GMm}{2E}