Orbital energetics

← Astro210

• total energy e is conserved \begin{list}{-}{}
• sum of K and U
• $E=K+U$
• $=\frac{1}{2}m{v}^2-\frac{GMm}{r}$

\item total E is conserved \item $E=(\frac{GMm}{L}{)}^2\frac{m}{2}(e^2-1)$ \item Hyperbolic orbit: $e>1,E>0,K>|U|$

• open orbit, unbound;, single perihelion passage at $\theta =0$

\item Parabolic orbit: $e=1,E=0,K=|U|$

• marginally unbound; velocity approach zero at infinite time

\item elliptical orbit: $e<1,E<0,K=|U|$ \item objects originating outside our solar system are easily identified by their total energy

• measure total energy (how far away it is, how fast is it moving)

\end{list}

Checking energy in circular orbits

• governing equation for circular orbits in scalar from
• $f=ma$
• $\frac{GM}{r^2}=\frac{v^2}{r}={\omega }^{2}r$
• $v=\sqrt{\frac{GM}{r}}$

Negative total energy orbits

• bound orbits have E < 0
• must add energy to break “unbind” the orbits

Parabolic orbits: escape speeds

• Escape speed is the speed that will bring your total pot energy to 0
• velocity becomes zero at infinite distance
• $\frac{1}{2}m{v}^2=\frac{GMm}{r}$
• $v_{esc}=\sqrt{\frac{2GM}{r}}$

Hohmann transfer orbit

• Elliptical transfer orbit from earth to superior planet \begin{list}{-}{}
• earths orbit becomes the transfer orbits perihelion passage
• inserted into superior planet orbit at aphelion. This constrains launch windows
• theoretically requires only two burns: at launch and aphelion insertion point

\item semimajor axis os transfer orbit \item $a_{to}=\frac{a+a_{sup}}{2};Earth$ \end{list}