Editing Spacecraft flight dynamics (section)

==Basic principles==
A [[space vehicle]]'s flight is determined by application of [[Isaac Newton|Newton]]'s [[second law of motion]]:
<math display="block">\mathbf{F} = m\mathbf{a},</math>
where '''F''' is the [[Euclidean vector|vector]] sum of all forces exerted on the vehicle, m is its current mass, and '''a''' is the acceleration vector, the instantaneous rate of change of velocity ('''v'''), which in turn is the instantaneous rate of change of displacement. Solving for '''a''', acceleration equals the force sum divided by mass. Acceleration is integrated over time to get velocity, and velocity is in turn integrated to get position.

Flight dynamics calculations are handled by computerized [[guidance system]]s aboard the vehicle; the status of the flight dynamics is monitored on the ground during powered maneuvers by a member of the [[flight controller]] team known in [[NASA]]'s [[Johnson Space Center|Human Spaceflight Center]] as the [[Flight controller#Flight dynamics officer (FDO or FIDO)|flight dynamics officer]], or in the [[European Space Agency]] as the spacecraft navigator.<ref>{{cite web |title=ESA - Flight Dynamics |url=https://www.esa.int/Enabling_Support/Operations/gse/Flight_Dynamics |publisher=European Space Agency |access-date=June 22, 2020}}</ref>

For powered atmospheric flight, the three main forces which act on a vehicle are [[propulsion|propulsive force]], [[aerodynamic force]], and [[gravity|gravitation]]. Other external forces such as [[centrifugal force]], [[Coriolis force]], and [[solar radiation pressure]] are generally insignificant due to the relatively short time of powered flight and small size of spacecraft, and may generally be neglected in simplified performance calculations.{{sfnp|Bate|Mueller|White| 1971| pp=11-12}}

===Propulsion===

The thrust of a [[rocket engine]], in the general case of operation in an atmosphere, is approximated by:<ref name="Sutton">{{cite book|author=George P. Sutton|author2=Oscar Biblarz|name-list-style=amp|title=Rocket Propulsion Elements|edition=7th|publisher=Wiley Interscience|date=2001|isbn=0-471-32642-9}} See Equation 2-14.</ref>

<math display="block">F = \dot{m}\;v_{e} = \dot{m}\;v_\text{e-opt} + A_{e}(p_{e} - p_\text{amb})</math>
where,
*<math>\dot{m}</math> is the exhaust gas mass flow
*<math>v_{e}</math> is the effective exhaust velocity (sometimes otherwise denoted as ''c'' in publications)
*<math>v_\text{e-opt}</math> is the effective jet velocity when ''p''<sub>amb</sub> = ''p''<sub>e</sub>
*<math>A_{e}</math> is the flow area at nozzle exit plane (or the plane where the jet leaves the nozzle if separated flow)
*<math>p_{e}</math> is the static pressure at nozzle exit plane
*<math>p_\text{amb}</math> is the ambient (or atmospheric) pressure

The effective exhaust velocity of the rocket propellant is proportional to the vacuum [[specific impulse]] and affected by the atmospheric pressure:<ref name="RPE7">{{cite book|first1=George P.|last1=Sutton|first2=Oscar|last2=Biblarz|title=Rocket Propulsion Elements|url=https://books.google.com/books?id=LQbDOxg3XZcC|publisher=John Wiley & Sons|date=2001|isbn=978-0-471-32642-7|access-date=28 May 2016|url-status=live|archive-url=https://web.archive.org/web/20140112033956/http://books.google.com/books?id=LQbDOxg3XZcC | archive-date=12 January 2014}}</ref>
<math display="block">v_e = g_0 \left(I_\text{sp-vac} - \frac{A_{e}\, p_\text{amb}}{\dot{m}}\right) </math>

where:
*<math>I_\text{sp-vac}</math> has units of seconds
*<math>g_0</math> is the gravitational acceleration at the surface of the Earth

The specific impulse relates the [[delta-v]] capacity to the quantity of propellant consumed according to the [[Tsiolkovsky rocket equation]]:<ref>{{cite book|author=George P. Sutton|author2=Oscar Biblarz|name-list-style=amp|title=Rocket Propulsion Elements| edition=7th| publisher=Wiley Interscience|date=2001|isbn=0-471-32642-9}} See Equation 3-33.</ref>
<math display="block">\Delta v\ = v_e \ln \frac {m_0} {m_1}</math>
where:
*<math>m_0</math> is the initial total mass, including propellant, in kg (or lb)
*<math>m_1</math> is the final total mass in kg (or lb)
*<math>v_e</math> is the effective exhaust velocity in m/s (or ft/s)
*<math>\Delta v </math> is the delta-v in m/s (or ft/s)

===Aerodynamic force===
[[Aerodynamic force]]s, present near a body with a significant atmosphere such as [[Earth]], [[Mars]] or [[Venus]], are analyzed as: [[lift (force)|lift]], defined as the force component perpendicular to the direction of flight (not necessarily upward to balance gravity, as for an airplane); and [[drag (physics)|drag]], the component parallel to, and in the opposite direction of flight. Lift and drag are modeled as the products of a coefficient times [[dynamic pressure]] acting on a reference area:{{sfnp|Anderson|2004| pp=257–261}}
<math display="block">\mathbf{L} = C_L q A_\text{ref}</math>
<math display="block">\mathbf{D} = C_D q A_\text{ref}</math>

where:
*''C''<sub>''L''</sub> is roughly linear with ''α'', the angle of attack between the vehicle axis and the direction of flight (up to a limiting value), and is 0 at ''α'' = 0 for an axisymmetric body;
*''C''<sub>''D''</sub> varies with ''α''<sup>2</sup>;
*''C''<sub>''L''</sub> and ''C''<sub>''D''</sub> vary with [[Reynolds number]] and [[Mach number]];
*''q'', the dynamic pressure, is equal to 1/2 ''ρv''<sup>2</sup>, where ''ρ'' is atmospheric density, modeled for Earth as a function of altitude in the [[International Standard Atmosphere]] (using an assumed temperature distribution, [[hydrostatic pressure]] variation, and the [[ideal gas law]]); and
*''A''<sub>ref</sub> is a characteristic area of the vehicle, such as cross-sectional area at the maximum diameter.

===Gravitation===
The gravitational force that a celestial body exerts on a space vehicle is modeled with the body and vehicle taken as point masses; the bodies (Earth, Moon, etc.) are simplified as spheres; and the mass of the vehicle is much smaller than the mass of the body so that its effect on the gravitational acceleration can be neglected. Therefore the gravitational force is calculated by:

<math display="block">\mathbf{W} = m \cdot g</math>

where:
*<math>W</math> is the gravitational force (weight);
*<math>m</math> is the space vehicle's mass; and
*<math>r</math> is the radial distance of the vehicle to the planet's center; and
*<math>r_0</math> is the radial distance from the planet's surface to its center; and
*<math>g_0</math> is the [[gravitational acceleration]] at the surface of the planet
*''g'' is the [[gravitational acceleration]] at altitude, which varies with the inverse square of the radial distance to the planet's center:{{sfnp|Kromis|1967| p=11:154}} <math display="block">g = g_0\left(\frac{r_0}r\right)^2</math>