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;