As a crude attempt to describe the velocity field of a galaxy with an oval distortion we have made a simple kinematical model, based on a suggestion by Dr. A. Toomre (note that this model was made in 1974, before more elaborate, and physically better founded, models were constructed by others. It should not be-taken too seriously; and is only meant as a guideline to determine the geometry of the system).
Consider a circular orbit at radius r in the plane of the
galaxy. We deform this circle into an ellipse by stretching it along the
x-axis and squeezing it along the y-axis. The Cartesian co-ordinates (x, y)
of a point P on the circle transform to (x’, y’) with:
x’ = x*f ; y’ = y/fa is the semi-major axis of the ellipse, b is the semi-minor axis. The circular velocity vector at P is (u, v) = (- Vc sin(phi) , Vc cos(phi) ), with phi = arctan y / x and Vc the amplitude of the circular velocity at radius r. We can eliminate phi and transform this vector also with equations (1) . Then we give the resulting ovals a constant pattern speed Omegap, but we reduce the angular velocity at each mean radius to compensate for this. We then have:
u’ = .... ; v’ = ....where r^2 = x^2f^2 + y^2/f^2 and Vc is taken at r.
mkdisk disk1 100000 rmax=2 mass=1
snapsquash disk1 disk2 1.1 1
snaprotate disk2 disk3 30,60 zx
snapgrid disk3 ccd3 moment=-1
ccdplot ccd3
or if you like NEMO’s pipe method, it can be done in one line:
mkdisk - 100000 rmax=2 mass=1 |\
snapsquash - - 1/1.1 1 |\
snaprotate - - 30,60 zy |\
snapgrid - bosma-vel.ccd moment=-1
ccdplot bosma-vel.ccd
nds9 bosma-vel.ccd
If you have ds9, the command nds9 will send this image into the ds9 display
server.
This can be compared in the same projection with the f=1 case, or now leaving out the programs and options we don’t need anymore:
mkdisk - 100000 rmax=2 mass=1 |\
snaprotate - - 60 y |\
snapgrid - bosma0-vel.ccd moment=-1
ccdplot bosma0-vel.ccd
nds9 bosma0-vel.ccd
28-Jul-06 V1.0 Created PJT/AB 23-oct-2021 adjusted example PJT