Alignment of the 2-m AST, which has an alt-azimuth mount, consists of aligning the aximuth axis vertically, making all the mechanical axes perpendicular to one another as much as possible, and colimating the mirrors so their axes coincide and are parallel to the mechanical axis of the tube.

1. ADJUSTMENT OF THE AZIMUTH AXIS: Place an accurate level on the drive ring between two attachment points and adjust the height of the leveling jacks to make the drive ring as level as possible. Repeat with another set of jacks, keeping one particular foot fixed until the base is leveled as well as possible. This procedure should level the base to about 0.5-1.0 arc min. This alignment is not overly critical, since an error corresponds to a compensating translation of the telescope in latitude and longitude. The effective longitude and latitude can be found by observing stars. The error should be kept as small as possible so the refraction calculation will work without defining two sets of coordinates.
2. ADJUSTMENT OF THE HEIGHT OF THE OIL PADS (HYDROSTATIC BEARING): The hydrostatic bearing pads are carried by spherical bearings on top of large acme screws that can be adjusted with a 3/8-inch square driver to change the heights of the pads. Set up a dial indicator attached to the fork with its quill on the top of the drive ring. Adjust the screws to give the same reading at all three pads. This makes the base of the fork parallel to the presumably level drive ring. It should be possible to make these levels the same to about 0.005 inches, or about 0.3 arc min. The measured runout among the pads (7/16/00) was 0.003 inches, with about 0.025 TIR around the drive circle due to variations of the height of the wheel.
3. MAKING THE TILT AXIS PERPENDICULAR TO THE AZIMUTH AXIS: Use the dial indicator with 2 in travel to measure the difference in level between the tops of the ends of the axles on opposite sides of the tube. Set up the indicator on one axle, then rotate the fork 180 degrees and measure the difference in level. Repeat for accuracy, and shim if necessary. Expect an uncompensated error of the order of 0.005 inches (or less) corresponding to about 8 arc sec. The actual error measured (7/20/00) was 0.009 inches, opposite the original error, or about 18 arcsec. We don't think the effort of correcting this is worth it.
4. MAKING THE TUBE PERPENDICULAR TO THE TILT AXIS: This involves using a plumb bob to define the axis as a vertical line. Set up the centering fixture in the top of the quadrapod with the plumb bob through it and hanging above the center mark in the auxiliary instrument head. Center the plumb bob on the mark to within better than 0.1 inches by shimming one of the side feet of the quadrapod. (The top and bottom feet will have to be shimmed half as much, of course.) Shimming a foot rotates the quadrapod and moves the top sideways by a similar amount. Be sure to rotate the fork 180 degrees in measuring the centering to be sure the centering error doesn't result from accumulated errors in orientations of the other axes. Expect a final uncompensated error in perpendicularity of this axis of about 0.1 inches in 100 inches, or 3 arc min. This is problematic in that the correction to the azimuth can be quite large at large zenith distance, roughly one degree for a misalignment of 3 arcmin at ZD=3 deg.
5. COLIMATING THE PRIMARY MIRROR: This is desirable because the primary mirror in the AST is very fast and must be evaluated at prime focus before tests begin with the sceondary mirror. It is a two-step process; first, define the mechanical axis of the tube with a light source, then autocolimate the primary mirror to that axis. Set up a special light source having a size of about 1/4 inch on the back wall of the enclosure about 2 inches farther from the vertex of the primary mirror (242 in) as the mirror's radius of curvature (240 in). The extra distance gives a smaller size of the horribly distorted refocused image in autocolimation. Use the unfocused image on the wall to point the telescope and move the wall to give a focused image. This setup can then be used to move the telescope under computer control to center a spot of light projected through a small hole centered in the centering jig on top of the quadrapod on the center mark of auxiliary instrument head. Next, use the three axial hard points, either manually or under computer control, to autocolimate the primary mirror to the mechanical axis. For adjusting the hard points with the motors, calculate the changes in height required, put a dial indicator on the shaft, and run the motor to give the desired change in position; the motors are too weak to run linearly, so don't count on calibrating their screws. The refocused image of the light source is a bright blob 0.25-0.5 inches in diameter surrounded by a halo about an inch in diameter. Given uncertainties in the centering of the primary mirror in its cell (0.05 in, or so) and the definition of the mechanical axis (0.05 inches from our optical technique), it should be possible to align the optical axis with the mechanical axis to about 2.5 arc min. The errors of the technique are as follows: centering of primary mirror, 1.5 arcmin; aligning the light source to the mechanical axis, 1.5 arcmin; autocolimating the mirror; 1.0 arcmin.
6. EVALUATING THE PRIMARY MIRROR: This involves observing a star; use the north star, which doesn't require any tracking to speak of. First, find the north star with the eyepiece, if not done already, then set up the CCD camera at the prime focus of the telescope. Second, find the north star and take an image of it with the CCD camera. Focus the camera with the computer-controlled motor provided to find the best focus. If necessary, move the telescope to minimize the coma in the image. Save the best image for evaluation of the primary mirror and its supports. Take a series of images on both sides of the focus to evaluate the structure of the wavefront. During tracking tests with the camera at prime focus, take pictures of other stars to test for problems with the mirror supports.
7. ALIGNING THE SECONDARY MIRROR: Place the secondary mirror into the telescope with its lifting jig, if it's not already there. Set up the CCD camera on the auxiliary instrument head at roughly the desired focus of the telescope. Point the telescope at the north star and move the secondary mirror to give a roughly focused image on the CCD. Move the secondary laterally and tilt it to eliminate coma at the center of the field of view. Translations and tilts of the secondary mirror will both produce coma in the final images. To first order, a translation of the secondary mirror (caused, for example, by miscolimation of the primary), may be compensated by a tilt of the secondary.
8. ECCENTRICITY OF THE DRIVE WHEELS: This is a simple measurement of the amplitude and zero point of the runout of the drive wheels. Values expected from measurements made in Nashville are 0.005 inches TIR for the azimuth axis and 0.001 inches TIR (or less) for the tilt axis. These values correspond to periodic pointing errors having an amplitude of 0.005/40 rad = 25 arcsec in azimuth, 4 arcsec in tilt. The absolute encoders, however, are likely to have larger values of such periodic errors, and we can actually see this effect in data collected for the tilt encoders. Another effect related to this is the temperature dependent gear ratio for tilt, which results from having an aluminum drive wheel and steel drive rollers. The different CTE's of these materials give roughly the following ratio: R = 1 + 5.1e-6DeltaT, with T in Fahrenheit. This corresponds to roughly 1500 counts over the full range of tilt and full range of temperature. The error for 10 deg F is then about 15 arcsec.

We will be using TPOINT, a standard program for assessing pointing errors of various telescopes, which we leased from Patrick Wallace in 1997. It allows assessment of various errors of interest in alt-az telescopes on the basis of a rational physical model of a generic mount. The effects we anticipate using are:

1. Misalignment of azimuth axis -- rotations about two axes. This involves correcting the azimuth and zenith distance for the two rotations with constants determined from TPOINT, where the correction is applied to the calculated coordinates and the constants ANS and AWE are in radians. The amplitude of this effect (displacement on the sky) is expected to be of the order of 1 arcmin in both axes. It may also be handled by assuming an effective (displaced) location of the telescope, as pointed out in no. 1 above. Actual measured values from close to 300 stars are about 15 arcsec north and 5 arcsec east.
   
   DeltaA = - ANS*sin(raz)/tan(rzd) - AWE*cos(raz)/tan(rzd)
   DeltaZD = ANS*cos(raz) - AWE*sin(raz)
   
   
2. Non-perpendicularity of tilt and azimuth axes (no. 3 above) -- one constant, which we'll measure mechanically. The maximum amplitude on the sky, near the zenith, is expected to be of the order of 0.3 arcmin. The expected value is 18 arcsec from measurements with a dial indicator; the measured value is 23 +/- 14 arcsec.
   
   DeltaA = NPAE/tan(rzd) = 9E-5/tan(rzd)
   
   
3. Non-perpendicularity of optical axis to tilt axis (Left-Right Collimation Error; no. 4 above) -- one constant, to be determined from stars. This is the dominant error for an alt-az telescope and is expected to give an error on the sky of appx. 3 arcmin. The value measured with TPOINT is 127 arcsec (2.1 arcmin), but this corresponds to a point somewhat off the optical axis.
   
   DeltaA = CA/sin(rzd)
   
4. Eccentricity of azimuth drive wheel - two constants. This effect gives a periodic error of one cycle per revolution of the azimuth axis with an semi-amplitude of the offset divided by the radius of the wheel, 0.005/40 = 1.25E-4 radians = 25 arcsec here, which gives the largest error on the sky near the horizon. The value measured with TPOINT is 29 arcsec for the incremental encoder and 3.5 arcmin for the absolute encoder. A related effect is eccentricity of the encoder mounts, which will give a periodic error with frequency of the gear ratio but diluted by the gear ratio at the main axis. The expected runouts will lead to errors of the order of 0.0005/2/20 = 3 arcsec for both azimuth and tilt.
   
   DeltaA = - ACES*sin(raz) - ACEC*cos(raz)
   
   
5. Eccentricity of tilt drive wheel - two constants. This effect gives a periodic error of one cycle per revolution of the tilt axis with an semi-amplitude of the offset divided by the radius of the wheel, 0.001/50 = 2E-5 radians = 4 arcsec here. The value measured with TPOINT is 92 arcsec (much bigger than we expected); the value for the tilt absolute encoder is a whopping 15 arcmin, which is probably affected by errors in other constants in the solution.
   
   DeltaZD = - ECES*sin(rzd) - ECEC*cos(rzd)
   
   
6. Sag of the tube - one constant. This error is indeterminate from shop measurements and will have to be measured by using stars. It is confused with eccentricity of the tilt drive wheel, which we don't expect to be important. The value measured with TPOINT is less than an arc second.
   
   DeltaZD = - TF*sin(rzd)
   
   
7. Zero point for azimuth - one constant. This constant is expected to be determined to several seconds of arc and to be stable unless there are wind-induced changes in the geometry of the tilt axis.
   
   DeltaA = - IA
   
   
8. Zero point for tilt - one constant. This value should be determinable to a few arcsec unless the tilt axis is unstable to wind-induced changes.
   
   DeltaZD = IE
   
   
9. Other, unexpected effects - four constants. In our first pointing experiments, we discovered a periodic error that is apparently related to a wave of about 0.006 inches in the surface of the azimuth bearing. This gives raw pointing errors of at least 1 arcmin, but these seem to be repeatable, and we have used TPOINT to quantify them. They have the form of
   
   DeltaA = - HASA3*sin(3*raz) - HACA3*cos(3*raz)
   DeltaZD = - HZSA3*sin(3*raz) - HZACA3*cos(3*raz)
   
   With the amplitudes being approximately 19 arcsec in azimuth and 39 arcsec in tilt.
   
   
10. A related effect that we must calibrate for this telescope is the temperature-dependent gear ratio for the tilt axis that results from using a steel drive cylinder on an aluminum wheel. The differences in the expansion coefficients of these materials give a roughly the following temperature dependence to the gear ratio:
    
    R = (1 + DeltaT*CTE_Al)/(1 + DeltaT*CTE_Steel)
    = 1 + DeltaT*(CTE_Al - CTE_Steel)
    = 1 + 5E-6*DeltaT per degree Fahrenheit,
    which gives approximately a 1500-count error (150 arcsec) over the full range of motion for a 100-deg maximum temperature range. For a 10-degree change in temperature, we thus expect a 15-arcsec error, which should be manageable, if we recalculate the gear ratio for temperature changes of the order of 5 degrees.
    

We have 16 constants to define, 14 from stars. Once the telescope is set up and functioning, the zero points should be stable, the eccentricities of the drive wheels should be stable, the three-theta errors should be stable, and the misalignment of the azimuth axis should be stable, so there will be effectively 2 constants (CA and, possibly, TF) to update as a task of routine maintenance. Both of these are related to instabilities in the tube, which could be expressed as seasonal changes in the geometry of the secondary-mirror supports, for example.

Corrections of stellar positions to raw positions of the telescope will be as simple as possible based on the model in TPOINT. We are using additive corrections to the two coordinates zd and AZ. These corrections should be updated often enough to keep them accurate, which means more often near the zenith. These update rates will be calculated so as to give errors less than 0.2-0.3 arcsec at the position of the field of the telescope. These rates are dominated by the terms for the nonperpendicularity of the optical axis to the tilt axis (CA, which is of the order of 3 arc min on the sky) and for the three-theta periodicity from the azimuth bearing. The first of these two effects dominates at small zenith distance, giving errors of up to 13 arcsec per minute. Update rates of 1 calculation per second should suffice to keep all these rates well within desired limits.

These procedures involve adding code to the control system to find stars in a list, centering them in the field of the telescope, calculating the instrumental positions of these stars, recording the expected and instrumental positions in a file that TPOINT will recognize, using TPOINT to determine the constants, transferring the constants to the header file for the telescope-control program (probably), and verifying the pointing model for the constants determined.