List of Figures:

Figure 1	Figure 2	Figure 3
Figure 4	Figure 5	Figure 6

View Figure 1 here

The science which can be carried out with the Two-Shooter and other large telescopes depends critically on the quality of the detected images. This image quality depends on the combination of atmosphere,telescope and instrument. Since many aspects of the telescope design, including cost, depend on the error tolerances, we must carefully consider the image quality which we expect to achieve. The goal adopted in the following error budget is that the telescope and its enclosure will degrade the image no more than the atmosphere alone in the best seeing. Most of the time, such a telescope will be entirely limited by the atmosphere, and may therefore appear over-designed. As Woolf and Angel (1980) pointed out in their design for MT-2, most of the science gets done in the best 10 -- 15% of the seeing (and other weather conditions). For photometric observations, the time to reach fixed signal-to-noise on a faint object against a background varies as the square of the image diameter. Interferometric studies benefit even more from improvements in image size . The need for compact images is especially great for a telescope such as the Two-Shooter, where we are ``expanding the envelope'' --- constantly pushing for better images of fainter objects.

When we specify our telescope design, we quickly run into some problems of definition and interpretation. The astronomer generally thinks about image size in terms of arcseconds. The engineer thinks about motion in inches or centimeters. The optician thinks about surface errors in microns. The accountant thinks about the budget in dollars. And, the atmosphere distorts the incoming wavefront based on a complex spectrum of turbulence. Once those differences in notation have been resolved, we must also consider the different, but not disparate, constraints imposed by various types of observations. Spectroscopy puts a premium on encircled energy to allow background rejection and resolution. Diffraction-limited imaging emphasizes modulation transfer function amplitude. Visible light imaging requires a compact, but not diffraction-limited, point spread function with low scattered light. The final result of the error budget will be to specify the detected point spread function. Depending on which of the above regimes is being considered, the image quality of a telescope may be specified in terms of a number of parameters. Some of these parameters and common units and abbreviations are described in Section 3.

2.1 Primary mirrors

2.2 Secondary complement

The largest of the three pairs of secondaries on the binocular telescope is the optical Cassegrain set. The fast Cassegrain focus has been optimized for wide field work in the optical and near-infrared wavelength regions. These mirrors form a Cassegrain focus at roughly F/5.17 without any additional optics. Coma limits this ``naked'' field-of-view to 2 arcminutes. Adding a small single element or doublet corrector can increase the field to around 5 arcminutes. An Epps' style 3-element refractive corrector with counter-rotating prisms to compensate atmospheric dispersion (ADC) will provide good images over a 50 arcminute field. The right side of Figure 2 shows the corrector optics mounted in the center hole of the primary. The focal ratio of the corrected field is F/5.4 with a platescale of 4.8 arcseconds per millimeter. The secondary mirror diameter is 1.93 meters with a sky baffle roughly 2.7 meters across. This mirror translates into the center section of the telescope on a trolley, and can thus be rapidly interchanged with the other secondaries which mount above it.

Infrared Cassegrain

The F/15 Cassegrain foci are optimized for thermal infrared observations requiring high throughput and the lowest possible background without the extra reflections of the combined focus. Two 0.71 meter diameter mirrors are made undersized to allow a fixed pupil against the sky when used in a chopping mode. The optical design allows a field-of-view up to 10 arcminutes at a platescale of 1.7 arcseconds per millimeter where the vignetting from the undersized secondary can be tolerated. These mirrors are interchanged with the F/33 secondaries with a flip-top mechanism that holds them on the same spiders. The classical Cassegrain F/15 optical configuration is shown on the left side of Figure 2. Tertiary mirrors can also be used to direct the light to a bent Cassegrain focal station.

View Figure 2 here

Combined Focus

The final set of secondaries is specifically optimized for interferometry using the full 22 meter baseline of the telescope. The F/33 secondaries provide a significantly longer back focal distance which permits us to bring the light from the two 8 meter primaries together at a combined focal plane with a platescale of 0.78 arcseconds per millimeter. The two 0.82 meter diameter secondaries are normally stored in the shadow above the F/15 secondaries. The tertiary mirrors which divert the light to the beam combiner flats are also moved into the beam on trolleys. Vignetting at the beam combiner limits the phased field of view to 6 arcminutes in diameter. This field corresponds to the size of the isoplanatic patch in the thermal infrared. The combined focus may be used with four reflections (including the primary) to reach an instrument mounted on the telescope elevation structure, or it may be used with six reflections to reach the gravity stable focus shown in Figure 3.

View Figure 3 here

3.1 Basic definitions for telescope images

• PSF() is the point spread function, or two-dimensional image profile as a function of angular radius, . The true profile is the convolution of diffraction, atmospheric distortion, telescope aberrations, and the original image. The point spread function will often be approximated by a gaussian to make combination of image spreads easier.

• FWHM (arcsec) is the full width at the half maximum intensity points in the image. This measure often used to characterize the image profile in a single number. Depending on the shape of the PSF, this may or may not be a useful calculation.

• RMS or root-mean-square diameter is also used to describe an image profile. RMS diameter is twice the standard deviation of the one-dimensional profile. For a gaussian image described by
  P(r,) = 1/(2²) * exp(-0.5r² / ²)

  the rms diameter is 2 . This is not the same as twice the rms distance from the center of the two-dimensional profile, 2* < r² > ^0.5, which is 2 larger. (This is the same distinction as the difference between speed and velocity in the kinetic theory of gases.) Both of these definitions seem to appear in the telescope literature with equal frequency. We shall use
  2 from now on.

• 80% ee is the image diameter which encircles 80% of the energy from a point source. For a gaussian profile, the 80% ee diameter is 3.59*. To refer to the alternate definition of rms diameter (2D, also called spotsize), 63% ee will be used.

• MTF, the modulation transfer function, describes the telescope and the atmosphere as an optical system. Throughput of the system is given as a function of spatial frequency. The modulation transfer function is a filter function which is applied to the Fourier transform of the incoming point spread function. As such, MTF is not easily expressed as a single number. The system MTF is the product of the MTFs of the individual components. For a typical telescope with a circular aperture, the MTF is circularly symmetric. Low spatial frequencies are transmitted efficiently, while spatial (or angular) frequencies near the diffraction limit are strongly attenuated. When the aperture is more complicated, like the phased two-shooter, the MTF becomes a two-dimensional function. See McCarthy, et. al. (1988) for some examples.

• S, the Strehl ratio, is the fractional degradation of the peak of the theoretical diffraction intensity. This is equivalent to the area under the MTF curve. The Strehl ratio is not often used in optical astronomy, but it is in fact buried in the definition of r₀. The Strehl ratio turns out to be useful in the infrared where the telescope becomes diffraction limited.

• r₀() (cm) is the coherence diameter of the atmosphere. Also known as Fried's length, r₀is the separation x for which the rms wavefront error from the atmosphere is 0.4175 wavelengths. r₀is often quoted for wavelength = 0.5 micron. Good seeing is indicated by a larger value of r₀. A telescope with diameter, D, larger than r₀will have an image size roughly equal to the diffraction pattern for diameter r₀. The value of r₀() scales as ^1.2.

3.2 Error budget strategy

The Structure Function Angel (1987) has outlined the error budget strategy for Columbus. Since our goal is to build a telescope which degrades the incoming image as little as possible, it seems appropriate to specify the errors in the telescope so they correspond to the distortions already induced by the atmosphere. The wavefront error induced by the atmosphere between two points separated by a distance x is given by

= 0.4175 * ( x / r₀ )^(5/6) waves

where the error is expressed as phase difference . No matter what value of r₀we use, the errors are always proportional to x^5/6 as long as the atmosphere retains a Kolmogorov spectrum. This allows us to relax the tolerance on the telescope optics and alignment at large scales since the atmosphere has already distorted the wavefront. We have adopted the ``structure function'' to describe the error in the incoming wavefront as a function of separation. Thus, by selecting a particular value of r₀, we may specify the permissible wavefront distortion induced by a mirror or a telescope. Converting from phase error to a linear dimension, we find the structure function:

² (x) = ( / 2)² 6.88 ( x / r₀) ^(5/3)

where (x) is the root mean square wavefront difference between points on the wavefront with spatial separation x. ( x, , r₀and should all have the same units.)

Telescope specifications

The first large mirror to use this sort of specification for polishing was the 4.2 m William Herschel Telescope (WHT) polished by Grubb-Parsons. The rms wavefront error was specified at 2, 8, 32 and 128 cm spacings. The specification is described in detail by Brown (1983) . Since then a number of other mirrors have been polished to specifications on several spatial scales.

Tilt Compensation

As we will see in later sections, various parts of the telescope contribute in different ways to the overall wavefront structure function. Tracking jitter introduces tilt of the wavefront or image motion. Alignment errors tend to produce low-order aberrations, while mirror support tends to introduce errors of higher spatial frequency. The large and small scale errors will be subdivided in slightly different ways. For example the primary mirror is not allowed to contribute any tilt to the wavefront since its axis defines where the telescope is pointing. Removing the mean tilt from the wavefront rolls off the structure function at large spatial scales as indicated by the correction factor shown here ¹¹.

² (x) = ( / 2)² 6.88 ( x / r₀) ^(5/3)[ 1 - 0.975 ( x / D) ^(1/3)]

D is the telescope aperture diameter. The tilt compensated structure function is shown in Figure 4.

Scattering Effects

The strict x ^5/6 power law is not maintained because diffraction effects allow us to relax the specifications at small spatial scales. Ruze (1966) gives the fractional loss due to scattering from small errors on scales much less than r₀as:

loss = (( 2) / ) ²

where is the rms deviation from the mean wavefront. We want to specify the small scale (1 cm) surface roughness so that no more than 5% of the light is scattered outside the seeing disk at 350 nm. = 12.5 nm implies a 17.6 nm rms wavefront error or a 9 nm rms surface error. The overall wavefront error budget can then be specified as the structure function:

² (x) = 2 ² + ( / 2)² 6.88 ( x / r₀) ^(5/3)

(x) is the rms wavefront difference between points separated by x.

Zenith Angle

When the telescope is looking away from the zenith through more of the atmosphere, we may expect the images to degrade. The seeing degrades according to (cos z) ^3/5, so an r₀= 45 cm atmosphere will be only r₀= 30 cm at a zenith angle, z, of 60° . We will allow the error budget to relax in this same fashion. Fixing at 12.5 nm and r₀at 45 ( / 0.5 µ m ) ^(6/5) ( cos z)^(-3/5) cm gives the numerical result of:

² (x,z) = 3.1*10^-16 + 1.65*10 ^-13 x^(5/3) cos z

for the structure function specification of the telescope error budget.

View Figure 4 here

3.3 Error budget translation guide

• 1.00 arcsec FWHM
• 0.85 arcsec 1D RMS diameter
• 1.00 arcsec 50% ee diameter
• 1.20 arcsec 2D RMS diameter
• 1.20 arcsec 63% ee diameter
• 1.52 arcsec 80% ee diameter
• 0.42 arcsec 1D image motion blur
• 10.1 cm r₀(0.5µ m) wavefront

3.4 Summary of the error budget

The error budget has been allocated among the various parts of the telescope according to the following criteria:

• How good can the atmosphere be? --- We will match it.
• How good can we do on telescopes today? --- We can do better.
• The project budget is minimized by allotting more error to expensive items.
• How much is left over? --- Divide it up.

• Tracking and Drives = 0.071 arcsec FWHM ~ 150 cm r₀
• Optical Alignment and Focus = 0.084 arcsec FWHM ~ 120 cm r₀
• Optical Design = 0.078 arcsec FWHM ~ 130 cm r₀
• Primary Mirror = 0.158 arcsec FWHM ~ 64 cm r₀
• Other Optical Surfaces = 0.063 arcsec FWHM ~ 160 cm r₀
• Telescope Seeing (5% ) = 0.056 arcsec FWHM ~ 180 cm r₀
• Atmospheric Seeing = 0.225 arcsec FWHM ~ 45 cm r₀
• Ideal Instrument = 0.080 arcsec FWHM ~ 126 cm r₀
• Actual Detected Image = 0.34 arcsec FWHM ~ 30 cm r₀Total

Once again, it must be emphasized that each of these numbers represents a more complicated spatial and temporal function. It will clearly be appropriate to review this specification, once the final numbers are in hand for the Mt. Graham Site Survey . The error budget details are being continuously iterated with the telescope design.

4.1 Strategy for tracking

4.2 Atmospheric image motion

= 0.043 * D^(-1/6) * r₀^(-5/6) arcsec

where D is the telescope diameter in meters, and r₀ ( 0.5 µ m) is also expressed in meters . For an 8 meter telescope in an r₀= 45 cm atmosphere, we expect 0.06 arcsec rms image motion. For the gaussian case, 0.06 arcsec rms motion would provide a 0.14 arcsec FWHM long-exposure image. Since 0.06 arcsec rms motion is derived by giving the entire error budget for wavefront tilt to the mount, we clearly must reach some compromises to allow for telescope -- telescope alignment and collimation errors. Unlike image size, image motion is constant with wavelength (neglecting diffraction), because atmospheric phase errors are independent of wavelength. Therefore, these tracking specifications should apply to all wavelengths. In the thermal infrared where the telescope becomes diffraction limited, we must consider the effects of image motion on the diffraction pattern. The minimum image size occurs around 5 microns where r₀approaches the size of a single 8-meter primary. To meet Marechal's criterion and preserve a Strehl ratio, S, of 0.8, again the whole error budget, image motion must remain smaller than 0.031 arcsec rms . This contributes a FWHM image size of 0.074 arcsec. To achieve S=0.95 it would be necessary to halve these numbers.

4.3 Guiding

After we have corrected pointing errors and drifts between the two telescopes, the next challenge for guiding is to remove the slowest parts of the atmospheric image motion to improve the long term image size. To estimate a timescale for the motion, we assume an outer turbulence scale of 100 meters and an upper atmosphere wind velocity of 20 meters/sec. This implies a timescale of up to 5 seconds. Image motions on timescales longer than one second should be correctable with normal guiding (moving the telescope) and tilting the secondaries. This allows us to loosen the tracking specification on longer timescales. The characteristic upper frequency of the atmospheric image motion is set by the pattern speed moving across the aperture. Using a 20 meter/sec wind and an 8 meter aperture we find a frequency of 2.5 Hz. Image motion and telescope motion on timescales longer than 0.1 second should be correctable with rapid guiding with a steering mirror if a sufficiently bright source is available within the isoplanatic region. A goal for rapid guiding would be to reduce net image motion below 0.01 arcsec rms --- actually improving on the atmosphere --- for frequencies below 10 Hz. Guiding at this level will certainly influence the design of the encoders and servo system, if not the telescope structure. Because the telescope aperture is so much larger than r₀, the improvements in image size from guiding are not as great as they would be for a smaller telescope. In the best seeing, we can expect less than a factor of two improvement. For optical imaging and spectroscopy, where the field-of-view may be much larger than the isoplanatic region in the focal plane, image motion across the field may not be correlated. Differential image motion should not be a problem in the thermal infrared, where the isoplanatic region is roughly the size of the focal plane.

4.4 Telescope pointing and tracking specification

The telescope should point to 0.3 arcsec rms at all times (night) with periodic recalibration of the open loop coefficients. The Multiple Mirror Telescope (MMT) already achieves this pointing performance . The telescope should track to 0.1 arcsec rms (1D) for periods up to 1000 seconds. The MMT currently tracks at 0.1 arcsec rms for shorter periods of time. This number also represents the blind offset specification for angular motion less than one degree. The telescope should track to 0.03 arcsec rms for periods up to 5 seconds. The value of 0.03 arcsec rms is somewhat larger than would be calculated from the r₀= 150 cm allowed in the wavefront error budget. This has been allowed because many of the other errors contribute less at large spatial scales. These tracking specifications should apply up to wind speeds of at least 6.7 m/sec (24 km/hour), and performance should degrade gracefully up to the maximum operating wind speed of 22 m/sec (80 km/hour) without exceeding three times the specification. The short timescale specifications imply an effective smoothness of a few microns for the drives and supports. Longer scale variations and temperature effects can presumably be taken out with encoders and look-up tables.

4.5 Telescope encoder specifications

The encoder specifications are derived from the telescope tracking specifications. We expect to have an encoder on the azimuth platform and one encoder on each of the two elevation C-rings. Additional encoders will be located on the instrument rotators etc.. We have dropped the 0.01 arcsec tracking requirement discussed previously, although we will still need 27 bit encoding to meet the 0.03 arcsec requirement.

4.6 Telescope drive specifications

The telescope shall be able to achieve a maximum angular velocity in each axis of 1.5 degrees/second. The maximum acceleration shall be 0.3 degrees/second ². These parameters and a 10 Hz telescope will allow a 1 arcminute offset in 0.6 seconds, a 1 degree offset in 4 seconds, and a 90 degree slew in 70 seconds. Motions less than about 7.5 degrees are acceleration limited . It is desirable to have a ``no-track'' cone no larger than 0.7 degree in diameter at the zenith. These velocity specifications are not intended to substantially impact the design or cost of a telescope which can meet the tracking goals.

4.7 Building drive specifications

The drive specifications for the co-rotating building enable it to follow the telescope and provide protection from foul weather.

5.1 Alignment strategy

5.2 Alignment error tolerances

5.3 Axial motion and focus

5.4 Lateral motion of the secondary

View Figure 5 here

5.5 Chopping and guiding

Some of the problems associated with vertex chopping of the secondary can be corrected by moving the pivot location. Lateral secondary misalignment can also be corrected by tilting the secondary to induce coma of the opposite sign. The secondary mirror generates no significant coma when it is rotated around its focal point near the prime focus. The problem, of course, is that other aberrations are generated. The largest aberration induced is astigmatism. Unlike configuration astigmatism which varies as the square of the field angle, the spotsize (in arcseconds) caused by alignment astigmatism varies linearly across the field. Empirically, this spotsize also varies inversely with the primary focal ratio ( F₁ ^(-1.75), for fixed vertex back focus) and linearly with system focal ratio (F_S, larger images in arcsec for slower Cassegrain foci). Presumably that is because the astigmatism is caused by the primary beam hitting the secondary asphere off axis, so the induced aberration varies with the secondary magnification. The spotsize increases as the square of the chop angle, but chop throws of 30 arcseconds are still possible. The tolerable chop throw increases linearly with the primary focal ratio (F₁). The penalty for this kind of chopping is the large increase in moment of inertia of the secondary around the pivot at the neutral point. Salinari shows that a faster primary actually decreases the moment of inertia for zero-coma chopping . In addition to astigmatism, rotating the secondary moves the entire curved focal surface. On-axis, the defocus caused by this motion overwhelms the astigmatism. For an F/4 Cassegrain focus with an F/1 primary, the secondary can rotate two degrees (0.6 degree image motion) without degrading the image on the axis of the secondary, while at the same time, the image in the original focal plane degrades to 10 arcsec or so. A large fraction of this spread can be recovered by refocusing and tilting the focal plane. To make things even more complicated, the induced astigmatism increases the curvature of the medial focal surface.

To summarize: it appears that chopping around the zero-coma or neutral point is feasible if we can deal with the moment of inertia problem. Some compromise in the pivot position also looks possible if the chop throw is restricted. Slow tilts of the secondaries can be used for telescope -- telescope coalignment over a range of a few arcminutes. Slow f/ratios are limited by induced astigmatism and field curvature, while wide fields are limited by focal plane tilts.

5.6 Specifications for the secondary motions

In detail, these specifications are derived by determining the amount of motion of the secondary mirror needed to generate an equivalent r₀ = 180 cm wavefront error or 0.03 arcsec of focal plane image motion (see the previous sections). The error allocation determined by that amount of secondary motion is divided by 10 to estimate the required resolution. Motions are assumed repeatable to four resolution elements. The tilt requirements have been calculated for the F/15 and F/33 secondaries. The secondary motions and resolutions for F/5 would be less than half as large, but we have chosen to keep the larger values for the specs and to gain some range (since the F/5 focus will not operate in a diffraction limited domain). The required range of motion depends on the mechanical adjustments in the telescope and the allowable motion which will maintain useful images. For example, a zero-coma chop motion of 1 arcminute requires 9 arcminutes of secondary tilt and 3 mm of translation for F/15.

5.7 Specifications for tertiary motions

Fine adjustments of the tertiaries are specified in the same way as the secondaries. Most of the tertiary motions are used to switch the beam from one focal station to another.

5.8 Corrector alignment

5.9 Vibration

6.1 Optical design

The wide field Cassegrain corrector has been separately specified to have images of 0.12 arcsec FWHM over the inner 30 arcminutes of field. Because the wavefront error budget increases as the telescope moves away from the zenith, the ADC is assumed to be ``off'' in the corrector specifications. The error allocation for optical design increases to 75 cm r₀at 60° zenith angle. Most of the increase can be used for the ADC design, so the full-on ADC can contribute 0.11 arcsec FWHM.

6.2 Primary mirrors

Category	0°	60°
Polishing	120 cm	120 cm
Optical Testing	270 cm	270 cm
Reflective Coating	360 cm	360 cm
Axial Support Distribution	180 cm	120 cm
Lateral Support Distribution	---	120 cm
Actuator Errors or Mass Distribution	180 cm	120 cm
Wind Forces on the Mirror	180 cm	90 cm
Ventilation Errors, * T	180 cm	180 cm
Homogeneity Errors, * T	180 cm	180 cm
Primary Mirror Total	64 cm	45 cm

Polishing and Testing

For fabrication of the primary mirror surface we have allocated a r₀= 120 cm wavefront for polishing and r₀= 270 cm for testing errors. Because the mirror will be polished on the axial supports, polishing can absorb another r₀ = 180 cm from the support budget provided the axial support pattern does not reverse itself to be worse than r₀= 120 cm away from the zenith (It doesn't). For spatial scales smaller than 10 cm, where diffraction (scattering) dominates atmospheric wavefront distortion, the surface roughness should be less than 6 nm rms.

The axis of the primary mirror is determined by the optical testing process and is constantly corrected by guiding. We have therefore removed tilt from the wavefront structure function that the figured surfaces are required to match. This substantially tightens the surface tolerances for large scale errors, but makes the tracking and alignment specifications more reasonable.

Asphere Tolerance

In addition to making a smooth optical surface, we must fabricate a particular conic section to be compatible with the optical design. At first look, we must produce the desired parabola to within the r₀= 180 cm tolerance. This absolute precision on the null lens or the radius measurement would be difficult or impossible to achieve. In practice, we would refocus the telescope or the corrector lenses to adjust to the conic section that is actually produced. As discussed in the focus section, spherical aberration limits the range of refocus to a few millimeters.

Focal Length Matching

The Columbus combined foci introduce an additional constraint, that of matching the focal lengths of the two telescopes. In order for the diffraction limited images at the edge of a combined field to overlap, the two telescopes must have the same platescale. We have specified that the images should overlap to within 0.1 / D at the edge of the isoplanatic patch. If we set = 10 / µ m D = 22 m and the isoplanatic patch to 6 arcminutes diameter, the images must align to 0.01 arcsec at the edge of the 3 arcminute radius field. This implies that the platescales must match to 1 part in 20000. Matching the focal lengths of the primaries to 0.5 mm sounds easy until we consider that the secondaries multiply this difference by a factor of 27.5 at F/33. Fortunately, we can refocus the telescope to adjust the scale and use the tertiary and beam combiner facets to adjust the pathlength. Increasing the primary focal length by 0.5 mm requires a change in secondary focal length of 0.07 mm to compensate, and the pathlength to the focal plane changes by 0.8 mm. Refocusing the telescope to move the focal plane 150 mm will adjust the platescale by 1 part in 20000. Based on this discussion, we should attempt to match the primary focal lengths to less than 0.25 mm.

Aluminizing

In order to convert the polished glass surface into a real mirror, we need to apply 100 nm of aluminum to the surface. Scaling results from smaller vacuum chambers and some numerical simulations suggest that we can apply a coating which is uniform to a few nm rms. A correspondingly small allocation of r₀= 360 nm has been added to the error budget. See Sabol, et. al. (1990) for more information on the Columbus aluminizing plans.

Mirror Support

When the telescope is pointing at the zenith, all of the gravity load is vertical and the mirror can be supported by axial actuators spread across the backplate. The number of axial supports is set so the mirror will not sag too much between the points of support, and so the support forces will not distort the surface. Studies of infinite repeating strips have shown that the optimal spacing of the supports turns out to be comparable to the thickness of the honeycomb structure. The axial actuators want to be located on the backplate so the forces spread out through the honeycomb before reaching the surface. 150 -- 200 axial support points are required to support the 8 meter F/1.2 blank with 85 cm edge thickness. This axial support configuration produce a 14 nm rms surface with a flat structure function which only approaches the r₀= 180 cm wavefront specification at spatial scales of 20 cm. The number of supports is, in fact, set by the small scale surface distortions. To avoid astigmatism which is the weakest bending mode of the mirror, systematic errors in the axial forces or parasitic forces from lateral actuators must be held below the 0.1% level.

As the telescope moves away from the zenith, the forces on the axial actuators decrease and the weight of the mirror is picked up by the lateral actuators. The ideal lateral support would apply forces parallel to the plane of the mirror at the local center-of-gravity of the blank. Applying forces in the center-of-gravity plane of a solid blank is difficult because you have to bore a hole to get there. The honeycomb structure has the opposite problem. There are plenty of holes, but you have to apply the forces to thin ribs of glass. By allowing the axial supports to apply a corrective force, we can apply all support forces from the backplate. A lateral support pattern with 100 actuators produces a 10 nm rms surface. This surface easily meets the r₀= 120 cm wavefront error allocation at 60° .

Wind Forces

Because 8 meter mirrors have such a large surface area, wind forces blowing on the mirror cannot be neglected. The locations of the actuators and the amplitudes of the support forces are optimized to handle the gravity load. Wind forces on the mirror will apply an additional force at approximately 2% of the gravity load. This much force (61 N m ^-2) pushing on the hardpoints alone would bend the mirror (400 nm rms) about three times more than the error budget will allow. Provided that the wind force is distributed among all the support points, a 12 m/sec wind blowing on the surface will not significantly distort the surface of the 8 meter F/1.2 honeycomb. We have specified that the telescope will meet the error budget in winds of 6.7 m/sec and will operate in winds up to 22 m/sec. No ``active'' adjustment of the mirror figure is required, because of the inherent stiffness of the honeycomb structure.

Thermal Control Tolerances

There are two major thermal effects on the primary mirror figure. Each of these has been allocated r₀= 180 cm in the error budget. First, the non-zero thermal expansion coefficient, = 3*10 ^-6 C°^-1, of borosilicate mirrors forces us to ventilate the primary honeycomb and control the temperature gradients across it. Finite element models of the F/1.2, 8 m mirrors have been used to investigate a number of thermal loading cases . We can stay within the 180 cm specification with a radial gradient of up to 0.25°C or a faceplate--honeycomb gradient of 0.30°C . Random temperature variations through the mirror need to be controlled to 0.15°C peak-to-valley. Cheng and Angel (1988) describe experiments which demonstrate this level of thermal control. Figure 6 shows the structure functions for these three thermal cases. Since the mirror will be figured in the laboratory, but operated at a temperature near 0°C , we also need to be concerned about variations in the expansion coefficient. This effect is common to both borosilicate and zero-expansion glasses. To meet the r₀= 180 cm wavefront, random variations in must be within 2*10 ^-8 C° ^-1 peak-to-valley. The glasses which were used in the 1.8 m and 3.5 m castings (Schott Tempax and Ohara E6) have roughly this variation.

The other consideration for thermal control is assuring that the mirror temperature tracks the ambient air temperature. If the front surface of the mirror is warmer than the local air, convective cells coming off the surface may degrade the local seeing. The rule of thumb is that a 1.0 °C temperature difference will introduce 0.3 to 0.5 arcsec of image degradation. See Woolf and Cheng (1988) for a review of experimental data and theoretical models. In order to keep the mirror seeing contribution below 0.06 arcsec, we require the faceplate temperature to be within roughly 0.15 °C of the ambient air. This constraint should also apply to other surfaces near the optical path of the telescope.

View Figure 6 here

6.3 Secondary mirrors and other optics

Wide Field Optical Secondary

The 2 m secondary mirror gets most of the error allocation because of its size. An r₀= 200 cm wavefront will scale to an equivalent r₀= 50 cm physical error on the actual mirror surface. The secondary error budget is distributed similar to that of the primary except that the error allowances for polishing and testing the convex surface have been increased to 90 cm and 135 cm respectively. Small scale surface errors should be less than 5 nm rms. The corrector fabrication will use the remaining 267 cm of wavefront error.

F/15 Chopping Secondary

Since the infrared Cassegrain focus has only two reflections and the wavefront is scaled from 8 m to 0.7 m, the chopping secondary can probably be fabricated and supported with a r₀= 300 cm wavefront (physically equivalent to r₀= 30 cm). The remaining 190 cm error can be contributed to the chopping motion which needs all the help it can get.

F/33 Combined Beams

Similarly, the F/33 secondary is specified to contribute less than r₀= 300 cm to the overall wavefront. The remaining errors will be divided among the tertiary flats and beam combiners.

1. P. A. Strittmatter, ``Columbus Project Telescope'', these proceedings.

2. R. G. Kron, et. al., Columbus Project Phase I Report 1988.

3. N. J. Woolf, and J. R. P. Angel, ``MT-2'', Steward Observatory Reprint # 259, 1980.

4. N. J. Woolf, ``High Resolution Imaging from the Ground'',
   Ann. Rev. Astr. Ap., 20, pp. 367-398, 1982.

5. J. R. P. Angel, W. B. Davison, J. M. Hill, E. Mannery, H. M. Martin,
   ``New Developments at the Steward Observatory Mirror Laboratory'', these proceedings.

6. J. M. Beckers, F. J. Roddier, P. R. Eisenhardt, L. E. Goad, and K-L. Shu,
   ``National Optical Astronomy Observatories (NOAO) Infrared Adaptive Optics Program I: general description'',
   Proc. S.P.I.E., 628, pp. 290-297, 1986.

7. D. W. McCarthy, E. K. Hege, J. D. Freeman, D. R. Blanco,
   J. C. Sjogren, C. C. Janes, J. W. Montgomery and S. B. Shaklan,
   ``Interferomtery with the Columbus Telescope: Design Considerations Based on
   MMT Experience and Imaging Simulations'',
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8. P. Dierickx, D. Enard, F. Merkle, L. Noethe, R. N. Wilson,
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9. J. R. P. Angel, ``Designing 8-m Mirrors for the Best Sites'',
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10. D. S. Brown, ``Optical Specification of Ground Based Telescopes'',
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11. H. M. Martin, private communication, 1989.

12. Ruze, J. 1966, Proc. I.E.E.E., 54, 633.

13. R. Cromwell, V. Haemmerle and N. Woolf,
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14. H. M. Martin, ``Image Motion as a Measure of Seeing Quality'',
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15. W. B. Wetherell,
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16. A. D. Poyner, J. W. Montgomery and B. L. Ulich,
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17. B. L. Ulich,
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    .

18. W. B. Wetherell and M. P. Rimmer,
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19. P. Salinari, private communication, 1986.

20. A. B. Meinel and M. P. Meinel,
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21. H. W. Epps,
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22. B. Atwood, private communication, 1988.

23. \sloppy H. W. Epps,
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24. F. Roddier,
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25. B. A. Sabol, B. Atwood, J. M. Hill, J. T. Williams, M. P. Lesser, P. L. Byard and W. B. Davison,
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26. G. Ballio, G. Parodi, P. Salinari, L. Fini, O. Citterio, R. Angel, L. Goble and J. Hill,
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27. A. Y. S. Cheng and J. R. P. Angel,
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28. N. J. Woolf and A. Y. S. Cheng,
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