# Linear Aerodynamic Terms#

In addition to the terms expressed previously, linear expressions are required for the aerodynamic angles and the total flightspeed.

## Angle of Attack#

Angle of attack is defined as

$\alpha\triangleq\arctan\frac{W}{U}$

which with the small perturbation theory is

$\alpha=\arctan\frac{W_0+w}{U_0+u}$

in stability axes, $$W_0=0$$

$\alpha=\arctan\frac{w}{U_0+u}$

and since $$w$$ is small

$\alpha\simeq\frac{w}{U_0+u}$

the perturbational forward speed is much smaller than the trim forward speed and the linear angle of attack is:

(82)#$\alpha=\frac{w}{U_0}$

## Sideslip#

Sideslip is defined as

$\beta\triangleq\arcsin\frac{V}{V_f}$

where $$V_f=\sqrt{U^2+V^2+W^2}$$. Looking at a linear expression for the total flightspeed:

\begin{split}\begin{align}V_f&=\sqrt{\left(U_0+u\right)^2+\left(V_0+v\right)^2+\left(W_0+w\right)^2}\\ &= \sqrt{\left(U_0+u\right)^2+v^2+w^2}\end{align}\end{split}

the trim $$U_0$$ is $$\gg$$ all the perturbational terms so

$V_f\simeq U_0$

giving the linear sideslip, subject to small $$v$$ as

$\beta=\frac{v}{U_0}$