The Heinrich-Hertz Telescope (HHT) is a 10 m radio telescope located on Mt Graham in Arizona, USA. Receivers are mounted at both Nasmyth foci. (A tertiary mirror which can be flipped 180 degrees determines which Nasmyth focus is being used.) The telescope is outfitted with both bolometers and heterodyne receivers.
The pointing history at the HHT is very short. In November of 1994, Wilhelm Altenhoff and Juergen Schmidt made the first pointing model fit to measurements they made with a 1.3 mm wavelength bolometer. They fit the 8-term pointing model used at the Effelsberg 100 m telescope to the data. The RMS residuals of their fits were 3.7-5.0 arcsec. This pointing model together with a table of additional (empirical) pointing corrections was then used until mid-1995. Since then there have only been a half-dozen full sky pointing measurements plus about a dozen other pointing sets with less than full sky coverage. Thus we are still learning about the pointing performance of the telescope and what can be done to improve it.
It is now realized that a ninth term must be added to the 100 m pointing model to describe the horizontal misalignment of a receiver at the Nasmyth focus. The nine terms arise from twelve sources of pointing errors likely to be present in the telescope. Since there are more sources than terms, one cannot measure the magnitude of the twelve from pointing measurements alone. Also, the terms are by no means orthogonal to each other. That is, some terms may have very similar effects upon the pointing corrections, especially in certain parts of the sky. To determine the coefficients of all the terms accurately, it is necessary to measure the telescope pointing errors over a wide distribution of points in the sky. The error sources are named below. For those familiar with the pointing model used at the Effelsberg, the corresponding ``P'' numbers are also given.
Name P# Description
NulA P1 Azimuth encoder zero point error
Col* P2 Collimation error in azimuth
Perp P3 Deviation of the azimuth and elevation axes from
orthogonality
IncW P4 Westerly tilt of the azimuth axis from the zenith
IncN P5 Northerly tilt of the azimuth axis from the zenith
P6 P6 No obvious physical effect. Would arise if the catalog
declinations of all objects were systematically too large.
NulE P7 Elevation encoder zero point error/elevation
collimation error
EFlx P8 Flexure in elevation proportional to the component of
gravity in the elevation direction.
ZFlx -- Flexure in elevation proportional to the component of
gravity along the axis of the primary. (Arises, for
example, if the subreflector is not supported at its
center of mass and therefore twists.)
OHor -- Horizontal offset of a receiver from the elevation axis
at the Nasmyth focus
OVer -- Vertical offset of a receiver from the elevation axis
at the Nasmyth focus
T0 -- Error in the sidereal time
The coefficients of the nine terms which can be determined by pointing measurements are:
Az Term El Term
NulA + OHor + T0*sin(latitude) cos(el) --
Col* 1. --
Perp - OVer sin(el) --
IncW - T0*cos(latitude) cos(az)*sin(el) -sin(az)
IncN sin(az)*sin(el) cos(az)
P6 sin(az) cos(az)*sin(el)
NulE -- 1.
EFlx + OVer -- cos(el)
ZFlx + OHor -- sin(el)
Since there are nine terms and twelve unknowns, one cannot solve for all twelve. One strategy would be to simply redefine nine new pointing constants equal to the above coefficients. However, it is mathematically equivalent to arbitrarily assume three of them are zero and solve for the remaining nine. We choose the latter option at the HHT. However, one must remember that the physical interpretation of these nine will be wrong if the three are not zero.
At the HHT, one bolometer is designated the ``prime'' bolometer, and a pointing fit is made assuming that T0, OHor, and OVer are zero. When another bolometer or receiver is used for observations, the first guess for the pointing model for it is that of the prime bolometer. If the receiver/ bolometer is at the same Nasmyth focus, it ought have the same pointing errors as the prime plus deviations due to OHor and OVer. A least squares fit for only OHor and OVer is then made to the pointing measurements after subtracting the prime model. The values of OHor and OVer so determined are not displacements of the new receiver/bolometer from the Nasmyth focus, but are displacements relative to the OHor and OVer of the prime bolometer which were assumed to be zero. When a receiver/bolometer is located at the opposite Nasmyth focus, its pointing model ought to be the prime model, plus the (relative) OHor and OVer, plus the error introduced by the misalignment of the tertiary mirror. This misalignment can be corrected by modifying the values Col* and NulE in the prime model. So one subtracts the prime model, then fits the remaining pointing errors by solving only for OHor, OVer, Col*, and NulE. (The Col* and NulE from this fit are then added to those of the prime model to produce the model for this receiver.)
One might ask why we do not solve for a complete pointing model independently for each receiver/bolometer. This would be a better method, ``all things being equal''. But things are never equal. The reason is sensitivity. A bolometer is more sensitive than a heterodyne receiver to continuum radiation. Thus there are more objects which can be quickly detected, and therefore one usually has better sky coverage for the pointing model fit. And the longest wavelength bolometer typically has the best sky coverage of all since the atmosphere is more transparent at longer wavelengths. The above strategy is needed only when there is not enough sky coverage for a full fit to the data from a non-prime receiver/bolometer.
Pointing measurements are almost exclusively made by scanning the telescope at a constant velocity in azimuth or elevation across an object. The intensity measured is plotted against position in the scan and a gaussian is fit in order to find the position of the peak and the beam width. Currently, we achieve the following accuracies at the HHT with the 1.3 mm bolometer:
Formal Errors from Gaussian Fits of Planets: 0.20-0.35 arcsec
RMS from repeated measurements of Planets: 0.7-1.1 arcsec
RMS residuals after full pointing model fit: 2-3 arcsec
(RMS errors before a fit to a new pointing model: 4-7 arcsec)
The planets have such high signal to noise that measurement errors due to bolometer/sky instabilities are negligible. The scatter in repeated measurements of planet positions is about three times greater than the accuracy of the gaussian fit. This is probably due to atmospheric (seeing) effects. We believe the telescope drive tracks smoothly to about 0.2 arcsec. At the shortest wavelength that the HHT will be used (0.35 mm), its beamwidth is 7 arcsec. The goal is therefore a pointing model with an RMS of 0.7-1 arcsec over the whole sky. The cause(s) for the actual RMS of 2-3 arcsec have not yet been determined.
The strategy of using a prime pointing model and only solving for the 2-4 terms which describe the offset of a receiver relative to the prime receiver has only been applied a few times. The results are mixed. Sometimes it seems to result in a good pointing model, other times it doesn't. We are confident that we can ultimately resolve the problem, but first we must find the cause. The following illustrates our recent experience. (The 1.3 mm and 0.9 mm bolometers were mounted at the same Nasmyth focus and the 345 SIS was at the opposite focus.)
Receiver Pointing Model Number of RMS residuals
Terms Fit Measures Az El
1.3 mm Bolometer All 9 34 1.4" 3.3"
(2 days later:)
0.9 mm Bolometer 2 Nasymth 24 2.3" 2.4"
(Adjusting above 0.9 mm by the fit and combining with the above 1.3 mm:)
1.3 mm \& 0.9 mm All 9 58 1.9" 2.9"
(1 week later:)
1.3 mm Bolometer All 9 49 1.6" 2.6"
(Adjusting 0.9 mm as above and combining:)
1.3 mm \& 0.9 mm All 9 73 2.0" 2.7"
(10 days later:)
345 SIS 4 Tert. \& Nas. 29+0 1.5" 4.6"
(3 days later:)
345 SIS 4 Tert. \& Nas. 21+7 2.9" 6.5"
(All 345 SIS combined:)
345 SIS 4 Tert. \& Nas. 50+7 2.9" 6.4"
345 SIS All 9 50+7 2.1" 2.8"
(Where two numbers are given for the number of measurements, the number made on
planets is given first followed by the number on objects at other declinations.)
The bolometer measurements demonstrate that the 0.9 mm measurements only differ from the 1.3 mm bolometer measurements by the Nasmyth offsets, and remain consistent with the 1.3 mm bolometer data even when combined with data taken a week later. But the 345 SIS measurements appear to differ by more than just the 4 offset terms. The fit of the 345 SIS data to all 9 terms indicates that the data are consistent with themselves and can be fit to the pointing model. But there were such large gaps in the sky coverage, that one would hesitate to depend upon the model in those gaps where it is unconstrained.
Examples of model fits to two different sets of 1.3 mm bolometer pointing measurements, and to the combined set follow:
Fit Name (A) (B) (C)
Date 8-9 Feb 96 17-18 Feb 96 8-9, 17-18 Feb 96
Model Constants:
NulA 348+-5" 373+-6" 370+-5"
Col* 4+-6" -11+-8" -22+-6"
Perp 0+-5" 14+-6" 23+-4"
IncW -2+-1" -8+-1" 5+-1"
IncN 10+-1" 11+-1" 12+-1"
P6 2+-1" 3+-1" 2+-1"
NulE -321+-7" -348+-8" -324+-6"
EFlx -25+-6" -14+-6" -28+-5"
ZFlx 10+-6" 36+-6" 18+-4"
N obs. 41 57 98
RMS of residuals:
Az El Az El Az El
8-9 Feb 1.6" 2.1" --- 1.4" 4.7"
17-18 Feb --- 2.9" 2.0" 3.3" 2.3"
8-9,17-18 Feb --- --- 2.7" 3.4"
(Two RMS pairs are listed for the 8-9 Feb and 17-18 Feb data sets. The second
is the RMS of the residuals in solution (C) for only those points.)
The combined data set results in a reasonable model fit, but the constants fitted don't always agree within their formal errors with the separate fits. For example, IncN and P6 are consistent for all three fits, but the others can differ by four times their formal error. Looking only at solutions (A) and (B), one might conclude that most of the sources of pointing error had changed in the eight days between the measurements. While some may have indeed changed, I suspect that most of the apparently significant changes are caused by small-number statistics combined with fact that some of the parameters of the fit are correlated with one another, and with the less than ideal sky coverage of the measurements. When two parameters are correlated, their combined effect on the pointing correction in a given area of the sky may be well determined, but their individual values may not. Small measurement errors may be magnified as regards their effect on the individual values. This ought to be reflected in the formal errors of these parameters, but small-number statistics may give unrealistic values for the estimates of these errors.