# Sideslip vs Freestream Azimuth Angle

Why is the sideslip angle, $$\beta$$, defined as

$$\beta=sin^{-1}\left(\frac{v}{|{\bf v}|} \right)$$

and not the same as freestream azimuth angle $$\psi$$

$$\psi=tan^{-1}\left(\frac{v}{u} \right)$$

where $${\bf v}=u \hat {\bf b}_x+v \hat {\bf b}_y+w \hat {\bf b}_z$$ is the airspeed of the aircraft in body coordinates, $$\hat {\bf b}$$, and $$|\cdot|$$ is euclidean norm of $$\bf v$$. $$\psi$$ can be interpreted as the angle between $$\hat {\bf b}_x$$ and the projection of $$\bf v$$ onto the bodies $$x$$-$$y$$ plane. What is the physical interpretation of $$\beta$$? • Is it helpful to consider the case of knife-edge flight, i.e. flight at a 90-degree bank angle, with the side of the fuselage supporting most of the aircraft weight? – quiet flyer Jun 19 '19 at 18:39
• @quietflyer, thank you, it is a thought provoking suggestion but I am struggling to gain any insight from it haha – eball Jun 19 '19 at 19:09
• Likewise I am not familiar enough with this highly mathematical approach to provide you a quick answer-- – quiet flyer Jun 19 '19 at 19:21
• Where did you find these definitions, please? – Cpt Reynolds Jun 19 '19 at 21:29
• For $\beta$ see PDF page 341: google.com/url?sa=t&source=web&rct=j&url=http://…. I don't have a reference for the freestream azimuth $\psi$...but it seems to be how most people think of sideslip. – eball Jun 20 '19 at 20:41

The sideslip angle, $$\beta$$, is the angle which the stability coordinate system needs to be rotated about the $$z$$ stability axis after being rotated by the angle of attack, $$\alpha$$, about the body $$y$$-axis. The sideslip angle is equal to the freestream azimuth angle $$\psi$$ when the angle of attack is zero.