Editing Spacecraft flight dynamics (section)

==Powered flight==
The equations of motion used to describe powered flight of a vehicle during launch can be as complex as six [[degrees of freedom (mechanics)|degrees of freedom]] for in-flight calculations, or as simple as two degrees of freedom for preliminary performance estimates. In-flight calculations will take [[perturbation (astronomy)|perturbation factors]] into account such as the Earth's [[oblateness]] and non-uniform mass distribution; and gravitational forces of all nearby bodies, including the Moon, Sun, and other planets. Preliminary estimates can make some simplifying assumptions: a spherical, uniform planet; the vehicle can be represented as a point mass; solution of the flight path presents a [[two-body problem]]; and the local flight path lies in a single plane) with reasonably small loss of accuracy.{{sfnp|Kromis|1967| p=11:154}}

[[File:Space launch flight diagram improved.png|thumb|upright|right|Velocity, position, and force vectors acting on a space vehicle during launch]]
The general case of a launch from Earth must take engine thrust, aerodynamic forces, and gravity into account. The acceleration equation can be reduced from vector to scalar form by resolving it into its tangential (speed <math>v</math>) and angular (flight path angle <math>\theta</math> relative to local vertical) time rate-of-change components relative to the launch pad. The two equations thus become:

<math display="block">\begin{align}
\dot{v} &= \frac{F\cos\alpha} m - \frac D m - g\cos\theta \\
\dot{\theta} &= \frac{ F\sin\alpha }{mv} + \frac L {mv} + \left( \frac g v - \frac v r \right) \sin\theta,
\end{align}</math>

where:
*''F'' is the engine thrust;
*''α'' is the angle of attack;
*''m'' is the vehicle's mass;
*''D'' is the vehicle's [[drag (physics)|aerodynamic drag]];
*''L'' is its [[lift (force)|aerodynamic lift]];
*''r'' is the radial distance to the planet's center; and
*''g'' is the [[gravitational acceleration]] at altitude.

Mass decreases as propellant is consumed and [[multistage rocket|rocket stages]], engines or tanks are shed (if applicable).

The planet-fixed values of v and θ at any time in the flight are then determined by [[numerical integration]] of the two rate equations from time zero (when both ''v'' and ''θ'' are 0):
<math display="block">\begin{align}
v &= \int_{t_0}^t \dot{v}\, dt \\
\theta &= \int_{t_0}^t \dot{\theta}\, dt
\end{align}</math>

[[Finite element analysis]] can be used to integrate the equations, by breaking the flight into small time increments.

For most [[launch vehicle]]s, relatively small levels of lift are generated, and a [[gravity turn]] is employed, depending mostly on the third term of the angle rate equation. At the moment of liftoff, when angle and velocity are both zero, the theta-dot equation is [[indeterminate form|mathematically indeterminate]] and cannot be evaluated until velocity becomes non-zero shortly after liftoff. But notice at this condition, the only force which can cause the vehicle to pitch over is the engine thrust acting at a non-zero angle of attack (first term) and perhaps a slight amount of lift (second term), until a non-zero pitch angle is attained. In the gravity turn, pitch-over is initiated by applying an increasing angle of attack (by means of [[gimbaled thrust|gimbaled engine thrust]]), followed by a gradual decrease in angle of attack through the remainder of the flight.{{sfnp|Kromis|1967| p=11:154}}{{sfnp|Glasstone|1965|p=209|loc=§4.97}}

Once velocity and flight path angle are known, altitude <math>h</math> and downrange distance <math>s</math> are computed as:{{sfnp|Kromis|1967| p=11:154}}

[[File:Gravity turn - landing - phase 2.svg|thumb|right|Velocity and force vectors acting on a space vehicle during powered descent and landing]]
<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>

The planet-fixed values of ''v'' and ''θ'' are converted to space-fixed (inertial) values with the following conversions:{{sfnp|Kromis| 1967|p=11:154}}
<math display="block">v_s = \sqrt{v^2 + 2\omega r v \cos\varphi \sin\theta \sin A_z + (\omega r \cos\theta)^2},</math>
where ''ω'' is the planet's rotational rate in radians per second, ''φ'' is the launch site latitude, and ''A''<sub>''z''</sub> is the launch [[azimuth]] angle.
<math display="block">\theta_s = \arccos\left(\frac{ v \cos\theta}{v_s} \right) </math>

Final ''v''<sub>''s''</sub>, ''θ''<sub>''s''</sub> and ''r'' must match the requirements of the target orbit as determined by orbital mechanics (see [[#Orbital flight|Orbital flight]], above), where final ''v''<sub>''s''</sub> is usually the required periapsis (or circular) velocity, and final ''θ''<sub>''s''</sub> is 90 degrees. A powered descent analysis would use the same procedure, with reverse boundary conditions.