Kepler's Equation

Kepler's Equation

• $$\begin{array}{rl}\vec{h}=& \left(x\dot{y}-\dot{x}y\right){\hat{i}}_h\\ =& \dots \\ \sqrt{\frac{\mu }{a^3}}=& \left(1-e\cos \left(E\right)\right)\frac{\mathrm{d}E}{\mathrm{d}t}\\ =& \dots \\ ⟹& \boxed{\sqrt{\frac{\mu }{a^3}}\left(t_1-t_0\right)=E-e\sin \left(E\right){|}_{E_0}^{E_1}}\end{array}$$

• What follows from Kepler’s Equation?

Period, Mean Angular Motion, Mean Anomaly

• One full rotation, $E_0=0\to E_f=2\pi$, will take one period: $P$
  • $$\begin{array}{rl}\sqrt{\frac{\mu }{a^3}}\Delta t& =E-e\sin \left(E\right){|}_{E_0}^{E_f}\\ \sqrt{\frac{\mu }{a^3}}P& =E-e\sin \left(E\right){|}_{E_0=0}^{E_f=2\pi }\\ \sqrt{\frac{\mu }{a^3}}P& =2\pi \end{array}$$
• This can be used to validate K3
• Mean Angular Motion:
  • $$\begin{array}{rl}n& \equiv \sqrt{\frac{\mu }{a^3}}\\ & =\frac{2\pi }{P}\end{array}$$
• Putting period and mean angular motion together:
  • $$n\left(t_1-t_0\right)=E-e\sin \left(E\right){|}_{E_0}^{E_f}$$
• Mean Anomaly:
  • Evolves linearly with time
  • $$M=n\Delta t+M_0$$

Reducing Kepler’s Equation

• $$\begin{array}{rl}n\underset{\text{input}}{\underbrace{\Delta t}}& =\left(E_1-e\sin \left(E_1\right)\right)-\underset{\text{const. based on θ0}}{\underbrace{\left(E_0-e\sin \left(E_0\right)\right)}}\\ n\Delta t+\left(E_0-e\sin \left(E_0\right)\right)& =E_1-e\sin \left(E_1\right)\\ n\Delta t+M_0& =E_1-e\sin \left(E_1\right)\\ ⟹& \boxed{M=E-e\sin \left(E\right)}\end{array}$$
• This is the reduced form of Kepler’s Equation
• The reduced form is non-linear $⟹$ no analytic solution

Solving the reduced form

• Newton invents new math again (Numerical Methods)

1. Create an error function:
   • $$\mathrm{f}\left(E\right)=M-\left(E-e\sin \left(E\right)\right)$$
   • If we guess $E$ correctly, $\mathrm{f}\left(E\right)=0$
     • $E^{\ast }\forall \mathrm{f}\left(E^{\ast }\right)=0$
     • Basically, we want the roots of $\mathrm{f}\left(E\right)$
2. Take a random value: $0\le E\le 2\pi$
3. Evaluate $\mathrm{f}\left(E\right)$ & $\frac{\mathrm{d}}{\mathrm{d}E}\left(\mathrm{f}\left(E\right)\right)$
4. Fit a line down to the x-crossing (when $y=0$)
   • Want $\mathrm{f}\left(E\right)+{\mathrm{f}}^{\prime }\left(E\right)\Delta E=0$
5. Solve for $\Delta E$:
   • $\Delta E=-\frac{\mathrm{f}\left(E\right)}{{\mathrm{f}}^{\prime }\left(E\right)}$
6. Repeat $\boxed{3},\boxed{4},\boxed{5}$ until $\Delta E<$ user-defined tolerance