Editing Spacecraft flight dynamics (section)

==Attitude control==
{{main|Attitude control}}

Since spacecraft spend most of their flight time coasting unpowered through the [[vacuum]] of space, they are unlike aircraft in that their flight trajectory is not determined by their attitude (orientation), except during atmospheric flight to control the forces of lift and drag, and during powered flight to align the thrust vector. Nonetheless, attitude control is often maintained in unpowered flight to keep the spacecraft in a fixed orientation for purposes of [[astronomical observation]], communications, or for [[solar power]] generation; or to place it into a controlled spin for passive [[spacecraft thermal control|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:{{sfnp|Beer | Johnston| 1972| p=499}}
<math display="block">\boldsymbol{\tau}_x = I_x \boldsymbol{\alpha}_x,</math>
where <math>\boldsymbol{\tau}_x</math> is the net [[torque]] about an axis of rotation exerted on the vehicle, ''I''<sub>x</sub> is its [[moment of inertia]] about that axis (a physical property that combines the mass and its distribution around the axis), and <math>\alpha_x</math> is the [[angular acceleration]] about that axis in radians per second per second. Therefore, the acceleration rate in degrees per second per second is
<math display="block">\boldsymbol{\alpha}_x = \tfrac{180}{\pi} \boldsymbol{\tau}_x/I_x,</math>

Analogous to linear motion, the angular rotation rate <math>\boldsymbol{\omega}_x</math> (degrees per second) is obtained by integrating '''α''' over time:
<math display="block">{\omega_x} = \int_{t_0}^t {\alpha_x} dt</math>
and the angular rotation <math>\boldsymbol{\theta}_x</math> is the time integral of the rate:
<math display="block">\theta_x = \int_{t_0}^t {\omega_x} dt</math>

The three principal moments of inertia ''I''<sub>x</sub>, ''I''<sub>y</sub>, and ''I''<sub>z</sub> 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 [[gimbaled thrust|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:
<math display="block">\boldsymbol{\tau} = \sum_{i=1}^N (\mathbf{r}_i \times \mathbf{F}_i ), </math>
where '''r''' is its distance from the center of mass, and '''F''' is the thrust of an individual thruster (only the component of '''F''' perpendicular to '''r''' is included.)

For situations where propellant consumption may be a problem (such as long-duration satellites or space stations), alternative means may be used to provide the control torque, such as [[reaction wheel]]s<ref>{{cite web|publisher=NASA |url=https://spinoff.nasa.gov/spinoff1997/t3.html |title=Reaction/Momentum Wheel |access-date=15 June 2018}}</ref> or [[control moment gyroscope]]s.<ref>{{cite journal|url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20100021932.pdf|title=Space Station Control Moment Gyroscope Lessons Learned|journal=Proceedings of the 40th Aerospace Mechanisms Symposium|date=12 May 2010 |last1=Gurrisi |first1=Charles |last2=Seidel |first2=Raymond |last3=Dickerson |first3=Scott |last4=Didziulis |first4=Stephen |last5=Frantz |first5=Peter |last6=Ferguson |first6=Kevin }}</ref>