Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag (when flying in the atmosphere of the Earth or other body, such as Mars or Venus).
The principles of flight dynamics are used to model a vehicle's powered flight during launch from the Earth; a spacecraft's orbital flight; maneuvers to change orbit; translunar and interplanetary flight; launch from and landing on a celestial body, with or without an atmosphere; entry through the atmosphere of the Earth or other celestial body; and attitude control. They are generally programmed into a vehicle's inertial navigation systems, and monitored on the ground by a member of the flight controller team known in NASA as the flight dynamics officer, or in the European Space Agency as the spacecraft navigator.
Flight dynamics depends on the disciplines of propulsion, aerodynamics, and astrodynamics (orbital mechanics and celestial mechanics). It cannot be reduced to simply attitude control; real spacecraft do not have steering wheels or tillers like airplanes or ships. Unlike the way fictional spaceships are portrayed, a spacecraft actually does not bank to turn in outer space, where its flight path depends strictly on the gravitational forces acting on it and the propulsive maneuvers applied.
Basic principles A space vehicle's flight is determined by application of Newton's second law of motion:
where F is the 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 systems 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 Human Spaceflight Center as the flight dynamics officer, or in the European Space Agency as the spacecraft navigator.
For powered atmospheric flight, the three main forces which act on a vehicle are propulsive force, aerodynamic force, and 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.
Propulsion
The thrust of a rocket engine, in the general case of operation in an atmosphere, is approximated by: See Equation 2-14.
where, is the exhaust gas mass flow is the effective exhaust velocity (sometimes otherwise denoted as c in publications) is the effective jet velocity when pamb = pe is the flow area at nozzle exit plane (or the plane where the jet leaves the nozzle if separated flow) is the static pressure at nozzle exit plane 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:
where: has units of seconds 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: See Equation 3-33.
where: is the initial total mass, including propellant, in kg (or lb) is the final total mass in kg (or lb) is the effective exhaust velocity in m/s (or ft/s) is the delta-v in m/s (or ft/s)
Aerodynamic force Aerodynamic forces, present near a body with a significant atmosphere such as Earth, Mars or Venus, are analyzed as: lift, defined as the force component perpendicular to the direction of flight (not necessarily upward to balance gravity, as for an airplane); and 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:
where: CL 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; CD varies with α2; CL and CD vary with Reynolds number and Mach number; q, the dynamic pressure, is equal to 1/2 ρv2, 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 Aref 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:
where: is the gravitational force (weight); is the space vehicle's mass; and is the radial distance of the vehicle to the planet's center; and is the radial distance from the planet's surface to its center; and 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:
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 for in-flight calculations, or as simple as two degrees of freedom for preliminary performance estimates. In-flight calculations will take 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.
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 ) and angular (flight path angle relative to local vertical) time rate-of-change components relative to the launch pad. The two equations thus become:
where: F is the engine thrust; α is the angle of attack; m is the vehicle's mass; D is the vehicle's aerodynamic drag; L is its 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 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):
Finite element analysis can be used to integrate the equations, by breaking the flight into small time increments.
For most launch vehicles, 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 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 engine thrust), followed by a gradual decrease in angle of attack through the remainder of the flight.
Once velocity and flight path angle are known, altitude and downrange distance are computed as:
The planet-fixed values of v and θ are converted to space-fixed (inertial) values with the following conversions:
where ω is the planet's rotational rate in radians per second, φ is the launch site latitude, and Az is the launch azimuth angle.
Final vs, θs and r must match the requirements of the target orbit as determined by orbital mechanics (see Orbital flight, above), where final vs is usually the required periapsis (or circular) velocity, and final θs is 90 degrees. A powered descent analysis would use the same procedure, with reverse boundary conditions.
Orbital flight
Orbital mechanics are used to calculate flight in orbit about a central body. For sufficiently high orbits (generally at least in the case of Earth), aerodynamic force may be assumed to be negligible for relatively short term missions (though a small amount of drag may be present which results in decay of orbital energy over longer periods of time.) When the central body's mass is much larger than the spacecraft, and other bodies are sufficiently far away, the solution of orbital trajectories can be treated as a two-body problem.
This can be shown to result in the trajectory being ideally a conic section (circle, ellipse, parabola or hyperbola) with the central body located at one focus. Orbital trajectories are either circles or ellipses; the parabolic trajectory represents first escape of the vehicle from the central body's gravitational field. Hyperbolic trajectories are escape trajectories with excess velocity, and will be covered under Interplanetary flight below.
Elliptical orbits are characterized by three elements. The semi-major axis a is the average of the radius at apoapsis and periapsis:
The eccentricity e can then be calculated for an ellipse, knowing the apses:
The time period for a complete orbit is dependent only on the semi-major axis, and is independent of eccentricity:
where is the standard gravitational parameter of the central body.
The orientation of the orbit in space is specified by three angles: The inclination i, of the orbital plane with the fundamental plane (this is usually a planet or moon's equatorial plane, or in the case of a solar orbit, the Earth's orbital plane around the Sun, known as the ecliptic.) Positive inclination is northward, while negative inclination is southward. The longitude of the ascending node Ω, measured in the fundamental plane counter-clockwise looking southward, from a reference direction (usually the vernal equinox) to the line where the spacecraft crosses this plane from south to north. (If inclination is zero, this angle is undefined and taken as 0.) The argument of periapsis ω, measured in the orbital plane counter-clockwise looking southward, from the ascending node to the periapsis. If the inclination is 0, there is no ascending node, so ω is measured from the reference direction. For a circular orbit, there is no periapsis, so ω is taken as 0.
The orbital plane is ideally constant, but is usually subject to small perturbations caused by planetary oblateness and the presence of other bodies.
The spacecraft's position in orbit is specified by the true anomaly, , an angle measured from the periapsis, or for a circular orbit, from the ascending node or reference direction. The semi-latus rectum, or radius at 90 degrees from periapsis, is:
The radius at any position in flight is:
and the velocity at that position is:
Types of orbit Circular