Computing the orientation of each cell w.r.t. the central pixel

We would like to keep the simulation cube at a distance d (element: X=0, YMAX/2, ZMAX/2 is at a distance d).

Now, we compute angle subtended by all cells, and compute differential rotation velocity as per Brand and Blitz (1991).

1) Read velocity cubes Vxx,Vyy,Vzz

a) Compute projects of components of V_rot (due to galactic rotation) along the los.

b) Compute projection of velocities Vxx, Vyy & Vzz along the line of sight (los)

c) V_los = V_rot + components from step a

Formulae

a)

tan(theta) = y / (d+x)

(D * D) = (d+x)^2 + (y*y)

R' = sqrt ( D^2 + R^2 + 2*R*D* cos(theta+L)

Compute V(R') using Brand & Blitz (1991)

R" = D sin(theta + L)

phi = 90 - (theta+L)

phi" = arccos(R"/R)

V_rot = V(R') * cos(90 - (phi+phi") ) = V(R') * cos(theta + L - phi)

b)

theta = arctan(y / (d+x) )

V"yy = Vyy cos(theta)

V"xx = Vxx sin(theta)

chi = arctan(z/ (d+x) )

V"zz = Vzz cos(chi)

c) v_los (x,y,z) = V"xx + V"yy + V"zz + V_rot

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