Editing Spacecraft flight dynamics (section)

==Translunar flight==
[[File:Tli.svg|thumb|200px|right|A typical translunar trajectory]]
Vehicles sent on lunar or planetary missions are generally not launched by direct injection to departure trajectory, but first put into a low Earth [[low Earth orbit|parking orbit]]; this allows the flexibility of a bigger [[launch window]] and more time for checking that the vehicle is in proper condition for the flight.

Escape velocity is not required for flight to the Moon; rather the vehicle's apogee is raised high enough to take it through a point where it enters the Moon's gravitational [[sphere of influence (astrodynamics)|sphere of influence]] (SOI). This is defined as the distance from a satellite at which its gravitational pull on a spacecraft equals that of its central body, which is
<math display="block">r_\text{SOI} = D\left(\frac{m_s}{m_c}\right)^{2/5},</math>
where ''D'' is the mean distance from the satellite to the central body, and ''m''<sub>''c''</sub> and ''m''<sub>''s''</sub> are the masses of the central body and satellite, respectively. This value is approximately {{convert|66300|km|nmi|sp=us|abbr=off}} from Earth's Moon.{{sfnp|Bate| Mueller | White| 1971| pp=333–334}}

An accurate solution of the trajectory requires treatment as a [[three-body problem]], but a preliminary estimate may be made using a [[patched conic approximation]] of orbits around the Earth and Moon, patched at the SOI point and taking into account the fact that the Moon is a revolving frame of reference around the Earth.

===Translunar injection===
{{Main|Translunar injection}}
This must be timed so that the Moon will be in position to capture the vehicle, and might be modeled to a first approximation as a Hohmann transfer. However, the rocket burn duration is usually long enough, and occurs during a sufficient change in flight path angle, that this is not very accurate. It must be modeled as a [[orbital maneuver#Low thrust for a long time|non-impulsive maneuver]], requiring [[numerical integration|integration]] by [[finite element analysis]] of the accelerations due to propulsive thrust and gravity to obtain velocity and flight path angle:{{sfnp|Kromis|1967| p=11:154}}
<math display="block">\begin{align}
\dot{v} &= \frac{F\cos\alpha}m - g\cos\theta\\
\dot{\theta} &= \frac{F\sin\alpha}{mv} + \left(\frac g v - \frac v r\right) \sin\theta, \\
v &= \int_{t_0}^t \dot{v}\, dt \\
\theta &= \int_{t_0}^t \dot{\theta}\, dt
\end{align}</math>
where:
*''F'' is the engine thrust;
*''α'' is the angle of attack;
*''m'' is the vehicle's mass;
*''r'' is the radial distance to the planet's center; and
*''g'' is the [[gravitational acceleration]], which varies with the inverse square of the radial distance:{{sfnp|Kromis|1967| p=11:154}} <math display="block">g = g_0\left(\frac{r_0}r\right)^2</math>

Altitude <math>h</math>, downrange distance <math>s</math>, and radial distance <math>r</math> from the center of the Earth are then computed as:{{sfnp|Kromis|1967| p=11:154}}
<math display="block">\begin{align}
h &= \int_{t_0}^t v \cos \theta\, dt \\
r &= r_0+h \\
s &= r_0 \int_{t_0}^t \frac v r \sin \theta\, dt
\end{align}</math>

===Mid-course corrections===
A simple lunar trajectory stays in one plane, resulting in lunar flyby or orbit within a small range of inclination to the Moon's equator. This also permits a "free return", in which the spacecraft would return to the appropriate position for reentry into the Earth's atmosphere if it were not injected into lunar orbit. Relatively small velocity changes are usually required to correct for trajectory errors. Such a trajectory was used for the [[Apollo 8]], [[Apollo 10]], [[Apollo 11]], and [[Apollo 12]] crewed lunar missions.

Greater flexibility in lunar orbital or landing site coverage (at greater angles of lunar inclination) can be obtained by performing a plane change maneuver mid-flight; however, this takes away the free-return option, as the new plane would take the spacecraft's emergency return trajectory away from the Earth's atmospheric re-entry point, and leave the spacecraft in a high Earth orbit. This type of trajectory was used for the last five Apollo missions (13 through 17).

===Lunar orbit insertion===
In the [[Apollo program]], the retrograde lunar orbit insertion burn was performed at an altitude of approximately {{convert|110|km|nmi|sp=us|abbr=off}} on the far side of the Moon. This became the pericynthion of the initial orbits, with an apocynthion on the order of {{convert|300|km|nmi|sp=us|abbr=off}}. The delta v was approximately {{convert|1000|m/s|ft/s|sp=us}}. Two orbits later, the orbit was circularized at {{convert|110|km|nmi|sp=us|abbr=off}}.<ref name="AFJ-LOI">{{cite web |last1=O'Brien |first1=Frank |title=Lunar Orbit Insertion |url=https://history.nasa.gov/afj/loiessay.html |website=Apollo Flight Journal |publisher=David Woods |access-date=June 25, 2020 |date=1999}}</ref> For each mission, the flight dynamics officer prepared 10 lunar orbit insertion solutions so the one could be chosen with the optimum (minimum) fuel burn and best met the mission requirements; this was uploaded to the spacecraft computer and had to be executed and monitored by the astronauts on the lunar far side, while they were out of radio contact with Earth.<ref name="AFJ-LOI"/>