Let dX and dY be the offsets on the sky parallel to the Azimuth axis and Elevation axis respectively, produced by a pointing error. The standard pointing model for the Plateau de Bure antennas has the following parameters:
IAZ azimuth encoder zero
dX = IAZ*cos(El) dY = 0
IEL elevation encoder zero
dX = 0 dY = IEL
COH telescope azimuth collimation
dX = cos(El)*asin(COH/cos(El)) dY = -asin(sin(El)/sqrt(1-COH**2))
for COH << cos(El), equivalent to
dX = COH dY = 0,
COV telescope vertical collimation
dX = 0 dY = COV
MVE Azimuth axis tilt towards East
dX = MVE*cos(Az)*sin(El) dY = -MVE*sin(Az)
MVN Azimuth axis tilt towards North
dX = -MVN*sin(Az)*sin(El) dY = -MVN*cos(Az)
NPE Elevation axis tilt (axis non perpendicularity)
dX = -NPE*sin(El) dY = 0
(assuming small NPE and COH in practice.)
REF0 First order refraction coefficient
dX = 0 dY = -REF0/tan(El)
REF1 Second order refraction coefficient
dX = 0 dY = -REF1/tan(El)**3
ELES gravity+eccentricity of Elevation encoder
dX = 0 dY = ELES*sin(El)
ELEC gravity+eccentricity of Elevation encoder
dX = 0 dY = ELEC*cos(El)
AZES eccentricity of Azimuth encoder
dX = AZES*sin(Az)*cos(El) dY = 0
AZEC eccentricity of Azimuth encoder
dX = AZEC*cos(Az)*cos(El) dY = 0
HEL Homology elevation bending (cos(El))
dX = 0 dY = -HEL*cos(El)
The total pointing correction is thus, in the small COH approximation
dX = IAZ.cos(El) + COH + sin(El) * (MVE*cos(Az)-MVN*sin(Az)-NPE)
+ cos(El) * (AZES*sin(Az)+AZEC*cos(Az))
dY = IEL+COV - (MVE*sin(Az)+MVN*cos(Az))
+ (ELES*sin(El)+(ELEC-HEL)*cos(El))
- REF0/tan(El) - REF1/tan(El)**3 - REF2/tan(El)**5
Since each antenna is equipped with an off-axis optical telescope, there are two sets of (COH,COV) values, one for the radio axis and one for the optical axis. As only IEL+COV is meaningful, the radio COV is set to zero, fixing the IEL value.
Five terms dominate the pointing model : IAZ, IEL+COV, COH, MVE and MVN. In principle COH, IEL, and COV depend only on the antenna, MVE and MVN depend on the station and antenna (azimuth bearing), and IAZ depends on the antenna (encoder), the station (footpad positions) and the positioning of the antenna on the station (repeatability 1 mm).
The refraction coefficients are weather dependant. Accordingly, the parameters are computed in real time, using temperature, pressure and humidity. REF0 and REF1 are thus fixed to zero in the pointing fit.
Among the other terms, all of which depend on the antenna, just two have been found to be of some importance for some antennas: ELEC which enters only in combination with HEL in radio (small values on all antennas), and possibly NPE (Antenna 3). These are marginally different from zero, however, and confirmation will be required. The other parameters can be set to zero.