# Horn positions, drift scan rotation angle.

The image on the left shows the 7 horns with the X axis along the azimuth arm. To do drift scans, you need to rotate the  feed array by an angle phi so that the projected centers on the X axis are equi-spaced. You would then place the azimuth at transit and let the sky drift by.
The beams on the sky are elliptical with the major axis along the azimuth arm. The spatial sampling on the sky along the X direction will be determined by the beam width (major axis) and the horn spacing (projected onto the X axis). The spatial sampling along the Y direction (ra) will be determined by the beam width (minor axis) and the temporal sampling rate of the data.

#### Solving for the drift scan rotation angle phi.

Let S be the distance from horn 0 (H0) center to the center of any of the other 6 horns. The starting horn angles are (the horns are spaced at 60 degree intervals):
H3 -30 degrees from X axis
H2 +30 degrees from the X axis
H1   0 degrees from the X axis

If we rotate the feed array by + phi degrees then the projection of each horn center on the X axis is:
Px1= S*sin(phi)
Px2=S*cos(phi+30)
Px3=S*cos(phi-30)

The separation between the projected horn centers are:
Distance(Px1-Px0)=S*sin(phi)
Distance(Px2-Px1)=S*(cos(30+phi)-sin(phi)=S*(cos(30)cos(phi)-sin(30)sin(phi)-sin(phi))

Equating Distance(Px1-Px0) with (Px2-Px1) gives:
sin(phi)=cos(30)cos(phi)-1.5*sin(phi)
2.5*tan(phi)=cos(30)
phi=atan(cos(30)/2.5)=19.1066 degrees

The projections onto the Y axis after rotation are:
Py1=S*cos(phi)
Py2=S*sin(phi+30)
Py3=S*sin(phi-30)

The position and offsets in units S of the horn separation is then (Positive Y is to the left to give a right handed system):

 Px,Py  (in units of horn separation S) (Px,Py)  S=26 cm| (Px,Py)  S=22cm Horn1 .327,.945 S (8.50,24.57) cm (7.19,20.79) cm Horn2 .655,-.756 S (17.03,-19.66) cm (14.41,-16.63) cm Horn3 .982,.189 S (25.53,4.91) cm (21.60,4.16) cm
• Fig 1 Top: separation in arc seconds along the projected X axis versus horn separation. The half beam width (216"/2) is over plotted in red.  To have the 3db point sit on the next beam requires a horn separation of 20.1 cm.
• Fig 1 2nd: Beam spacing vs horn separation in units of .5*(FWHM) . A 26 cm horn separation is 1.3 times larger than the halfbeamwidth sampling.
• Fig 1 3rd: This is the beam spacing the in the orthogonal direction versus horn displacement.
• Fig 1 bottom: This shows the time difference for the same RA to transit in different horns.

• These calculations have not taken into account all of the asymmetries in the focal plane (e.g. X scale factor not constant).
processing: x101/mbeam/mbhorns/doit.pro

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