Mathematical Description
The simplest case of the gravity turn trajectory is that which describes a point mass vehicle, in a uniform gravitational field, neglecting air resistance. The thrust force is a vector whose magnitude is a function of time and whose direction can be varied at will. Under these assumptions the differential equation of motion is given by:
Here is a unit vector in the vertical direction and is the instantaneous vehicle mass. By constraining the thrust vector to point parallel to the velocity and separating the equation of motion into components parallel to and those perpendicular to we arrive at the following system:
Here the current thrust to weight ratio has been denoted by and the current angle between the velocity vector and the vertical by . This results in a coupled system of equations which can be integrated to obtain the trajectory. However, for all but the simplest case of constant over the entire flight, the equations cannot be solved analytically and must be integrated numerically.
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