Astrodynamics/Time of Flight

Astrodynamics

Eccentric Anomaly

The eccentric anomaly, E, is the angle measured at the geometric center of the orbit between the periapsis and the projection of the satellite position on an auxiliary circle of radius a.

Because the position of an orbiting body is not directly related to its true anomaly additional angular parameters must be introduced to determine the time of flight. The eccentric anomaly is defined as:

$\tan {\frac {E}{2}}={\sqrt {\frac {1-e}{1+e}}}\tan {\frac {\nu }{2}}$

The trajectory equation in terms of eccentric anomaly is:

$r=a(1-e\cos E)$

The mean anomaly is the angle that an orbiting body would travel in a certain time if it were in a circular orbit whose radius is the semi-major axis of its orbit. Recall the formula for mean motion:

$M=nt=t{\sqrt {\frac {\mu }{a^{3}}}}$

The relation between eccentric and mean anomaly is expressed in Kepler's equation:

$M=E-e\sin E$

Time of Flight

From Kepler's Equation, the time since periapsis is:

$t={\sqrt {\frac {a^{3}}{\mu }}}(E-e\sin E)$

More generally:

$t-t_{0}={\sqrt {\frac {a^{3}}{\mu }}}(E-E_{0}-e\sin(E-E_{0}))$

Non-Elliptical Trajectories

Hyperbolic Trajectory

Kepler's equation for a hyperbolic trajectory is:

$M=e\sinh H-H$

Where H is the hyperbolic anomaly and is analogous to eccentric anomaly:

$\tanh {\frac {H}{2}}={\sqrt {\frac {e-1}{e+1}}}\tan {\frac {\nu }{2}}$

Note that these equations make use of hyperbolic sine and tangent functions.

Parabolic Trajectory

The time since periapsis for a parabolic trajectory can be expressed via Barker's Equation:

$t={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D+{\frac {D^{3}}{3}}\right)$

Where the parabolic anomaly, D is:

$D=\tan {\frac {\nu }{2}}$