Editing Laplace–Runge–Lenz vector (section)

== Derivation of the Kepler orbits ==
[[File:Laplace Runge Lenz vector2.svg|thumb|right|350px|Figure 2: Simplified version of Figure 1, defining the angle {{mvar|θ}} between {{math|'''A'''}} and {{math|'''r'''}} at one point of the orbit.]]

The ''shape'' and ''orientation'' of the orbits can be determined from the LRL vector as follows.<ref name="goldstein_1980" /> Taking the dot product of {{math|'''A'''}} with the position vector {{math|'''r'''}} gives the equation
<math display="block">
\mathbf{A} \cdot \mathbf{r} = A \cdot r \cdot \cos\theta = 
\mathbf{r} \cdot \left( \mathbf{p} \times \mathbf{L} \right) - mkr,
</math>
where {{mvar|θ}} is the angle between {{math|'''r'''}} and {{math|'''A'''}} (Figure 2). Permuting the [[triple product|scalar triple product]] yields
<math display="block">
\mathbf{r} \cdot\left(\mathbf{p}\times \mathbf{L}\right) = 
\left(\mathbf{r} \times \mathbf{p}\right)\cdot\mathbf{L} = 
\mathbf{L}\cdot\mathbf{L}=L^2
</math>

Rearranging yields the solution for the Kepler equation
{{Equation box 1
|indent =:
|equation = <math>
\frac{1}{r} = \frac{mk}{L^2} + \frac{A}{L^{2}} \cos\theta</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}

This corresponds to the formula for a conic section of [[eccentricity (orbit)|eccentricity]] ''e''
<math display="block">
\frac{1}{r} = C \cdot \left( 1 + e \cdot \cos\theta \right)
</math>
where the eccentricity <math>e = \frac{A}{\left| mk \right|} \geq 0</math> and {{mvar|C}} is a constant.<ref name="goldstein_1980" />

Taking the dot product of {{math|'''A'''}} with itself yields an equation involving the total energy {{mvar|E}},<ref name="goldstein_1980" />
<math display="block">
A^2 = m^2 k^2 + 2 m E L^2,
</math>
which may be rewritten in terms of the eccentricity,<ref name="goldstein_1980" />
<math display="block">
e^{2} = 1 + \frac{2L^2}{mk^2}E.
</math>

Thus, if the energy {{mvar|E}} is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"<ref name="goldstein_1980" />), the eccentricity is greater than one and the orbit is a [[hyperbola]].<ref name="goldstein_1980" /> Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a [[parabola]].<ref name="goldstein_1980" /> In all cases, the direction of {{math|'''A'''}} lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach.<ref name="goldstein_1980" />