Editing Spacecraft flight dynamics (section)

===Hyperbolic departure===
The required hyperbolic excess velocity ''v''<sub>∞</sub> (sometimes called ''characteristic velocity'') is the difference between the transfer orbit's departure speed and the departure planet's heliocentric orbital speed. Once this is determined, the injection velocity relative to the departure planet at periapsis is:{{sfnp|Bate| Mueller| White| 1971| p=369}}
<math display="block">v_p = \sqrt{\frac{2\mu}{r_p} + v_\infty^2}\,</math>

The excess velocity vector for a hyperbola is displaced from the periapsis tangent by a characteristic angle, therefore the periapsis injection burn must lead the planetary departure point by the same angle:{{sfnp|Bate| Mueller| White| 1971| p=371}}
<math display="block">\delta = \arcsin\frac 1 e</math>

The geometric equation for eccentricity of an ellipse cannot be used for a hyperbola. But the eccentricity can be calculated from dynamics formulations as:{{sfnp|Bate| Mueller| White| 1971| p=372}}
<math display="block">e = \sqrt{1+\frac{2\varepsilon h^2}{\mu^2}},</math>
where {{mvar|h}} is the specific angular momentum as given above in the [[#Flight path angle|Orbital flight]] section, calculated at the periapsis:{{sfnp|Bate| Mueller| White| 1971| p=371}}
<math display="block">h = r_p v_p,</math>
and ''ε'' is the specific energy:{{sfnp|Bate| Mueller| White| 1971| p=371}}
<math display="block">\varepsilon = \frac{v^2}2 - \frac \mu r\,</math>

Also, the equations for r and v given in [[#Orbital flight|Orbital flight]] depend on the semi-major axis, and thus are unusable for an escape trajectory. But setting radius at periapsis equal to the r equation at zero 
anomaly gives an alternate expression for the semi-latus rectum:
<math display="block">p = r_p(1 + e),\,</math>
which gives a more general equation for radius versus anomaly which is usable at any eccentricity:
<math display="block">r = \frac{r_p(1 + e)}{1+e\cos\nu}\,</math>

Substituting the alternate expression for p also gives an alternate expression for a (which is defined for a hyperbola, but no longer represents the semi-major axis). This gives an equation for velocity versus radius which is likewise usable at any eccentricity:
<math display="block">v = \sqrt{\mu\left (\frac{2}{r}-\frac{1-e^2}{r_p(1+e)}\right)}\,</math>

The equations for flight path angle and anomaly versus time given in [[#Flight path angle|Orbital flight]] are also usable for hyperbolic trajectories.