Editing Orbital mechanics (section)

===Kepler's equation===
One approach to calculating orbits (mainly used historically) is to use [[Kepler's equation]]:
:<math> M = E - \epsilon \cdot \sin E </math>.

where ''M'' is the [[mean anomaly]], ''E'' is the [[eccentric anomaly]], and <math> \epsilon </math> is the [[eccentricity (mathematics)|eccentricity]].

With Kepler's formula, finding the time-of-flight to reach an angle ([[true anomaly]]) of <math>\theta</math> from [[periapsis]] is broken into two steps:
# Compute the eccentric anomaly <math>E</math> from true anomaly <math>\theta</math>
# Compute the time-of-flight <math>t</math> from the eccentric anomaly <math>E</math>

Finding the eccentric anomaly at a given time ([[Kepler's Equation#Inverse problem|the inverse problem]]) is more difficult. Kepler's equation is [[transcendental function|transcendental]] in <math>E</math>, meaning it cannot be solved for <math>E</math> [[algebraic function|algebraically]]. Kepler's equation can be solved for <math>E</math> [[analytic function|analytically]] by inversion.

A solution of Kepler's equation, valid for all real values of <math> \textstyle \epsilon </math> is:

<math display="block">
E =
\begin{cases}

\displaystyle \sum_{n=1}^{\infty}
{\frac{M^{\frac{n}{3}}}{n!}} \lim_{\theta \to 0} \left(
\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}} \left[ \left(
\frac{\theta}{ \sqrt[3]{\theta - \sin(\theta)} } \right) ^n \right]
\right)
,  & \epsilon = 1  \\

\displaystyle \sum_{n=1}^{\infty}
{ \frac{ M^n }{ n! } }
\lim_{\theta \to 0} \left(
\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}} \left[ \left(
\frac{ \theta }{ \theta - \epsilon \cdot \sin(\theta)} \right) ^n \right]
\right)
, &  \epsilon \ne  1

\end{cases}
</math>

Evaluating this yields:

<math display="block">
E =
\begin{cases} \displaystyle
x + \frac{1}{60} x^3 + \frac{1}{1400}x^5 + \frac{1}{25200}x^7 + \frac{43}{17248000}x^9 + \frac{ 1213}{7207200000 }x^{11} +
\frac{151439}{12713500800000 }x^{13} \cdots \ | \ x = ( 6 M )^\frac{1}{3}
,  & \epsilon = 1  \\
\\
\displaystyle
\frac{1}{1-\epsilon} M
- \frac{\epsilon}{( 1-\epsilon)^4 } \frac{M^3}{3!}
+ \frac{(9 \epsilon^2 + \epsilon)}{(1-\epsilon)^7 } \frac{M^5}{5!}
- \frac{(225 \epsilon^3 + 54 \epsilon^2 + \epsilon ) }{(1-\epsilon)^{10} } \frac{M^7}{7!}
+ \frac{ (11025\epsilon^4 + 4131 \epsilon^3 + 243 \epsilon^2 + \epsilon ) }{(1-\epsilon)^{13} } \frac{M^9}{9!} \cdots

, &  \epsilon \ne  1

\end{cases}
</math>

<br/>Alternatively, Kepler's Equation can be solved numerically.  First one must guess a value of <math>E</math> and solve for time-of-flight; then adjust <math>E</math> as necessary to bring the computed time-of-flight closer to the desired value until the required precision is achieved.  Usually, [[Newton's method]] is used to achieve relatively fast convergence.

The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits.  For near-parabolic orbits, eccentricity <math>\epsilon</math> is nearly 1, and substituting <math>e = 1</math> into the formula for mean anomaly, <math>E - \sin E</math>, we find ourselves subtracting two nearly-equal values, and accuracy suffers.  For near-circular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all).  Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it does not hold for parabolic or hyperbolic orbits.  These difficulties are what led to the development of the [[universal variable formulation]], described below.