8.1 Primary Mirror Support Systems Against Gravity 8.1.1 Introduction
Foucault solved the problem of maintaining a mirror with a high reflective coating by introducing glass mirrors, which then can be easily re-silvered after chemical removal of the tarnished coating. This ended the era of speculum metal mirrors which required, when tarnished, a repolishing within a seeing limited - or diffraction limited - criterion. The chemical process was later replaced by the vacuum deposition process by J. Strong (cf. Sect. 1.1.5).
The next problems for the primary mirror of large telescopes were to minimize their elastic deflection under gravity - or own weight deformation - and also that resulting from thermal gradients. This latter problem was appropriately solved by the inventions and developments of low expansion materials such as Pyrex, Sitall, fused silica, vitrocerams, and silicon carbide.
The design of axial and lateral supports for large telescope mirrors necessarily act onto several mirror subareas which support a small fraction of the mirror weight. In addition to some reference subareas, an appropriate number of additional subareas are distributed all over the back surface and possibly on the edge surface of the mirror. Whatever the geometrical distribution where these forces act, the associated system of forces must be non-hyperstatic. In other terms, the set of the supporting forces must be in astatic equilibrium so the 3D-orientation of the mirror remains unchanged with respect to the reference zones.
Three small reference areas, generally located at 120° and near the mirror edge, allow us to determine its axial position whilst two or three small reference areas are required to define the lateral position. The number of these latter areas depends on whether the telescope is an equatorial mount or alt-azimuthal mount.
G.R. Lemaitre, Astronomical Optics and Elasticity Theory, Astronomy and Astrophysics Library, DOI 10.1007/978-3-540-68905-8.8, @ Springer-Verlag Berlin Heidelberg 2009
Let P denote the weight of the mirror, Na the number of axial supports and Nl the number of lateral supports. We shall hereafter refer to passive axial and lateral support systems as astatic systems that deliver reaction sets of equal forces fa and mean forces < f > in the axial and lateral directions respectively; thus N fl + Nf < fl >2= P2 and these forces are fa = (P/Na) cosz, < fl >=(P/Nl) sinZ, (8.1)
where z is the zenith distance angle of the telescope axis.
In such passive support systems, the assumption of invariance of the mirror orientation is strictly valid only if the flexure of the cell structure remains axisymmetric in all orientations of the telescope. We note here that the telescope star-pointing error resulting from deviation from this hypothesis is easier to compensate for an altazimuth mount than with an equatorial mount. These corrections are achieved by the telescope control system.
Several types of passive astatic systems have been invented for axial support (Fig. 8.1) and lateral support (Fig. 8.2).
Fig. 8.1 Passive astatic systems for the axial support of large mirrors. (A) Articulated stacked segments - ridges, triangles, or both - invented by T. Grubb  and known as whiffletree assembly. (B) Folded astatic levers invented by Lassel . (C) Open-loop air pressure cushions introduced by Foucault . (D) Open-loop hydraulic pads
The mirror support system with articulated stacked segments - or whiffletree assembly - was introduced by T. Grubb  for the axial support of the 1.8-m Lord Rosse telescope. Because of the difficulty to obtain such stiff and lightweight systems, these devices are limited to thin mirrors whose aperture diameter does not exceed 2 m, hence preventing resonance instability problems that are generated from wind buffeting. For instance each 1.8-m hexagon segment mirror of the Keck telescopes (t/d = 1/24) is supported by three whiffletrees acting on 3 x 12 = 36 pads,
Fig. 8.2 Passive astatic systems for the lateral support of large mirrors. (E) Direct astatic levers acting on rear apertures or on edge. (F) Perimeter bag filled with liquid of the same mirror density or narrow mercury bag. (G) Open-loop radial hydraulic pads acting on edge
Fig. 8.2 Passive astatic systems for the lateral support of large mirrors. (E) Direct astatic levers acting on rear apertures or on edge. (F) Perimeter bag filled with liquid of the same mirror density or narrow mercury bag. (G) Open-loop radial hydraulic pads acting on edge which for the 36 segments of a telescope gives Na = 1,296 pads . Within this mirror size, the other classic alternative uses folded astatic levers.
For monolithic mirror sizes in the range 4-8 m, although an air pressure system is the "perfect" system, the most common passive axial support systems are astatic levers or controlled hydraulic pads linked together in each of three sectors. An option combining folded and direct astatic levers in a single device was adopted for the 5 m Hale telescope mirror (see the description by Bowen ) and 6-m Sao telescope mirror as emphasized by Balega . The option with an air pressure cushion was developed by Stepp et al. for the 8-m primary mirrors of the Gemini telescopes [79, 80] in addition to an active control system that incorporates Na = 120 hydraulic pads distributed on five concentric rings. With its other alternative of partial vacuum, it turns out that this latter option is sometimes preferred for the supporting of smaller mirrors such as Cassegrain mirrors (for instance, see Bely  for Cfht). The lateral support system of 8 m mirrors alternatively employs direct astatic levers or hydraulic pads.
Various schools of thought progressively led to noticeably differing solutions for the mirror geometry and the mirror weight per unit surface area. For instance, let us consider some existing large telescope mirrors (cf. focal-ratios in Table 1.1) with their aspect-ratios t/d and the Na, Nl pad-numbers of their associated support systems.
• Moderately lightweight ribbed mirrors: A typical case of a moderately lightweight mirror cast with a ribbed structure is the 5-m aperture mirror of the Hale telescope at Palomar [9, 10]. The plane-concave blank in Pyrex (aspect-ratio t/d = 1/8.33, Na = Nl = 36 pads) includes cylindrical and equally spaced cavities where the axial and lateral astatic support acts. These open cavities are 36 in number and distributed in a 3-fold symmetry where the ribs converge. All ribs and the mirror surface are 10-12 cm thick. Very elaborated and precise astatic support systems, 33 in number, designed in a double astatic form, provide in a single device both axial and lateral supporting forces within an accuracy of 0.2%.
The primary mirror of the 6 m telescope of the Special Astrophysical Observatory in Caucasus is a meniscus of 65 cm thickness (t/d = 1/9.3) with Na = Nl = 60 combined axial-lateral astatic levers installed in 60 holes of 31 cm diameter and 43 cm depth that are distributed along four concentric rings. From chief designer B.K. Ioannisiani, the mirror is made from the Pyrex type glass known in Ussr as "glass No. 316" whose coefficient of thermal expansion is a = 3 ± 0.3 x 10~6 [6, 31].1
• Thick solid mirrors: A classical case of a thick solid mirror is the 4 m mirror of the Mayall-Kpno telescope at Kitt Peak (t/d = 1/8, Na = 36) with large support
1 The 6-m telescope of the Special Astrophysical Observatory was the first large telescope erected with an alt-az mount, a concept which was followed up for all subsequent large telescopes.
pads along two concentric rings. This geometry with flat rear side was adopted for many telescope primaries in the 4 m class.
• Sandwich honeycomb mirrors: The spin casting technique applied to sandwich honeycomb mirrors provided the two 8.4 m mirror blanks of the Lbt at Mount Graham. The two Pyrex borosilicate mirrors are plano-convex with 0.9 m thickness at the edge (aspect-ratio t/d = 1/9.3, Na — 418) and the lateral support system, described by Parodi et al., mainly acts on the rear faceplate  (Fig. 8.3).
• Thin solid meniscus mirrors: The spin casting technique allowed production of 8.2 m menicus blanks for the main four primary mirrors of the Vlt at Cerro Paranal . The Zerodur vitoceram blanks were cast by Schott and figured into a thin shape (t/d = 1/47). The axial support system allows an active control of the mirror shape by use of 147 hydraulic systems. These and the three reference positions act on the mirror via 150 articulated tripods (Na = 450 pads) distributed over six concentric rings. The lateral support system is achieved by radial hydraulic pads surrounding the mirror  (Fig. 8.3). Similar 8-m thin meniscus mirrors were cast in fused silica by Corning for the primaries of Gemini-1, -2  and Subaru .
8.2 Density and Thermal Constants of Mirror Substrates
Several important features enter into the choice of a mirror substrate. Beside the ability to obtain a fine surface polish and easily renew the reflective coating, we may briefly list some of them: large mechanical rigidity, linear strain-response for active optics shaping, low total mass, high resonant frequency of the fundamental vibration mode against wind buffeting, low deformation to environmental thermal change.
Some important constants in the choice of a mirror substrate are the following:
• Stiffness: The flexure of a mirror supported into gravity and its resonance frequency of the fundamental mode from wind buffeting are functions of the Young modulus E to the density ^ of the material. Given a material, several merit-ratios in the form Ewhere m e [0, 1, 3/2, 2] may characterize a stiffness (see Sect. 8.4.1) depending on whether comparisons are with equal volume, equal mass, external or internal bending forces. The higher the stiffness, the smaller the flexure and the higher the natural resonance frequency of the fundamental mode.
• Coefficient of thermal expansion a (CTE): The coefficient of thermal expansion is the dimensional response of a material to temperature change.
• Thermal diffusivity dt. This constant enters in all differential equations on heat propagation through a solid; it is defined as2
^Cp and characterizes how quickly a thermal equilibrium is achieved.
• Specific heat Cp: The specific heat is the heat energy required to raise the temperature of a given amount of a substance by one degree. For a solid substance it is typically measured under constant pressure and expressed in [JKg-1 K_1].
• Thermal conductivity kt: the thermal conductivity is the intensive property of a material that indicates its ability to conduct heat. It is defined as the quantity of heat, Q, transmitted in time t through a thickness L, in a direction normal to a surface of area A, due to a temperature difference A T, under steady state conditions and when the heat transfer depends only on the temperature gradient. The thermal conductivity = heat flow rate x distance / (area x temperature difference): kt = Q x ALAT, expressed in [Wm-1K-1].
Quantities kt and dt are both time dependent. Quantities ^, a, and dt are given in Table 8.1.
2 A. Couder introduced the reciprocal of the thermal diffusivity dt for astronomical mirrors, which he also checked with several experiments, and listed the numerical values a/dt for various glass, Pyrex, and metal substrates (, p. 308). Since, after a thermal disturbance, metal mirrors recover equilibrium much more quickly than glass or Pyrex mirrors, Couder pointed out that a thin metal mirror would be almost completely insensitive to thermal shocks. Thus, he proposed a drum-shaped mirror design - or vase form where a thin meniscus is surrounded by a ring - and experimented with some of them in cast iron. Couder further developed such mirrors with a 20 cm aperture and enamel deposits, and obtained interesting results.
D. Maksutov, apparently without knowledge of Couder's table, quite similarly established a thermo-mechanical table for the merit features of mirror materials. Also arriving at the conclusion that metal mirrors are almost insensitive to thermal shocks, he developed and built several stainless steel mirrors the largest of them was ~80cm in diameter for his 70-cm aperture wide-field telescope at Pulkovo Observatory .
ß [103 Kg/m3]
Beryllium pure VHP
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