spin for passive thermal control, or to create artificial gravity inside the craft. Attitude control is maintained with respect to an inertial frame of reference or another entity (the celestial sphere, certain fields, nearby objects, etc.). The attitude of a craft is described by angles relative to three mutually perpendicular axes of rotation, referred to as roll, pitch, and yaw. Orientation can be determined by calibration using an external guidance system, such as determining the angles to a reference star or the Sun, then internally monitored using an inertial system of mechanical or optical gyroscopes. Orientation is a vector quantity described by three angles for the instantaneous direction, and the instantaneous rates of roll in all three axes of rotation. The aspect of control implies both awareness of the instantaneous orientation and rates of roll and the ability to change the roll rates to assume a new orientation using either a reaction control system or other means. Newton's second law, applied to rotational rather than linear motion, becomes: where is the net torque about an axis of rotation exerted on the vehicle, Ix is its moment of inertia about that axis (a physical property that combines the mass and its distribution around the axis), and is the angular acceleration about that axis in radians per second per second. Therefore, the acceleration rate in degrees per second per second is Analogous to linear motion, the angular rotation rate (degrees per second) is obtained by integrating α over time: and the angular rotation is the time integral of the rate: The three principal moments of inertia Ix, Iy, and Iz about the roll, pitch and yaw axes, are determined through the vehicle's center of mass. The control torque for a launch vehicle is sometimes provided aerodynamically by movable fins, and usually by mounting the engines on gimbals to vector the thrust around the center of mass. Torque is frequently applied to spacecraft, operating absent aerodynamic forces, by a reaction control system, a set of thrusters located about the vehicle. The thrusters are fired, either manually or under automatic guidance control, in short bursts to achieve the desired rate of rotation, and then fired in the opposite direction to halt rotation at the desired position. The torque about a specific axis is: where r is its distance from the center of mass, and F is the thrust of an individual thruster (only the