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An experimental 2.7 meter liquid mirror telescope Gibson, Bradley Kenneth 1990

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A N E X P E R I M E N T A L 2.7 M E T E R L I Q U I D M I R R O R T E L E S C O P E B y B R A D L E Y K E N N E T H G I B S O N B . S c , Universi ty of Waterloo, 1988 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Geophysics of Astronomy We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A September 1990 © B r a d l e y Kenneth Gibson, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Geophysics & Astronomy The University of British Columbia Vancouver, Canada Date September 17, 1990 DE-6 (2/88) Abstract A 2.7-meter L i q u i d Mi r ro r Telescope ( L M T ) is currently under construction at the Universi ty of B r i t i s h Columbia . The stationary, parabolic pr imary mirror is formed by uniformly rotating the highly reflective l iquid , metallic mercury. Compensation for the lack of mechanical tracking wi l l be accomplished by using the T ime Delay and Integrate ( T D I ) readout technique wi th our Ford 2048 x 2048 C C D detector. The abil i ty to produce large, diffraction-limited mirrors in the laboratory has been previously demonstrated; this project, the first of its k ind , is an investigation into their potential for astronomical survey-work, in a working observatory environment. A set of 40 intermediate-band filters, one filter to be used per "photometric" night, w i l l facilitate the collection of Spectral Energy Distr ibutions (SEDs) of a l l objects, to a l imi t ing stellar V m a s ~ 2 1 , in the 83.0 deg 2 strip (1/3 deg wide) available to this telescope. A catalog of >10 5 galaxy and ~3,000 quasar S E D s is expected, providing the largest database of its type to date. A detailed stress-tensor analysis of the mercury-loaded mirror cell is given. The max imum flexure of the cell (at the mirror r im) was found to be ^0.32 m m . The high resonant frequency of the cell was designed to minimise the excitement of both gravity and capillary waves, surface phenomena which can degrade image quality. A n analysis of the support structure indicated that its max imum deflection under wind loading ( ^ 0 . 3 / m i ) would be significantly less than the Ford C C D ' s physical pixel size of 15 /J.m. Temperature-sensitive autofocussing was needed to ensure that thermal ex-pansion/contraction of the support structure d id not lead to defocussing. Star-trai l curvature at non-zero latitudes and the discrete nature of the T D I readout mode leads to elongation of north-south (NS) and east-west ( E W ) image structure, respectively. Convolving stellar Point Spread Functions (e.g. Gaussian) wi th the C C D ' s pixel width showed image broadening of ~ 5 % ( E W ) and ~ 9 % (NS) . W h i l e these effects are ex-pected to be negligible for our instrument, quantifying them under on-site testing wi l l be imperative before proceeding wi th the development of larger L M T s . i i Table of Contents A B S T R A C T i i L I S T O F T A B L E S v i L I S T O F F I G U R E S v i i A C K N O W L E D G E M E N T S v i i i 1. L I Q U I D M I R R O R T E L E S C O P E S 1 1.1 Introduction 1 1.2 History 3 1.2.1 1850 - 1872 3 1.2.2 1909 - 1922 4 1.2.3 1982 - Present 6 1.3 Summary 9 2. L I Q U I D M I R R O R T E L E S C O P E S : G E N E R A L C O N C E P T S 10 2.1 Introduction 10 2.2 Basics 10 2.2.1 The Surface of a Rota t ing F l u i d 10 2.2.2 L i q u i d Mercury as a Mi r ro r 12 2.2.3 D a t a Acquis i t ion 13 2.3 Complicat ions 15 2.3.1 Curvature of the Ea r th 15 2.3.2 Rota t ion of the Ea r th 17 2.3.3 A x i s Misalignment 18 2.3.4 Surface Waves 20 2.3.5 T D I C C D Readout Mode 25 2.3.6 Star Tra i l Curvature 28 i i i 2.4 The Laval Experiments 31 2.5 Summary 37 3. U B C / L A V A L 2.7m L M T : D E S I G N A N D S T R U C T U R A L A N A L Y S I S . . . 38 3.1 Introduction 38 3.2 M i r r o r 38 3.2.1 M i r r o r Ce l l 38 Composi t ion 39 Flexure 40 Dis tor t ion 46 Longi tudinal Stabil i ty 47 3.2.2 L i q u i d Mercury Surface 49 3.2.3 M i r r o r Support 51 3.3 Instrumentation Support Structure 51 3.3.1 Resonant Frequency 52 3.3.2 W i n d Loading 54 3.3.3 Thermal Effects 55 3.4 Instrumentation 56 3.4.1 Detector 56 3.4.2 Corrector Lenses 57 3.4.3 Fil ters 58 3.4.4 Focussing Mechanism 58 3.5 Opt ica l Testing 59 3.6 Summary 60 iv 4. U B C / L A V A L 2.7m L M T : S C I E N T I F I C P R O G R A M 61 4.1 Introduction 61 4.2 Telescope Specifications 61 4.3 The Survey 63 4.3.1 General Characteristics 63 4.3.2 L i m i t i n g Magnitudes and Number Counts 64 Galaxies 64 Quasars and Stars 66 4.3.3 Spectrophotometric Redshifts 67 4.4 The Impact of the L M T on Astronomy 69 4.4.1 Ga laxy Spectral Evolu t ion 69 4.4.2 Cosmology 70 4.4.3 Large Scale Structure 71 4.4.4 Structure of the M i l k y Way 72 4.5 Summary 74 R E F E R E N C E S 108 A P P E N D I X A : M I R R O R C E L L S T R E S S A N A L Y S I S 118 v List o f Tables Table I: K e y Contributions i n the Development of the L M T 75 Table II: U B C / L a v a l 2.7m L M T : Site and M i r r o r Specifications 77 Table III: U B C / L a v a l 2.7m L M T : Detector Specifications 78 Table I V : Pr ime Focus Corrector 79 Table V : Fil ters 80 Table V I : Galaxies W i t h Published Redshifts i n the L M T Strip (Epoch 1991.0) 81 Table V I I : Galaxies W i t h Publ ished Redshifts i n the L M T Strip (Epoch 2001.0) 83 Table V I I I : Ga laxy Clusters i n the L M T Strip (Epoch 1991.0) 85 Table I X : Galaxy Clusters i n the L M T Strip (Epoch 2001.0) 87 Table X : Quasars i n the L M T Strip (Epoch 1991.0) 89 Table X I : Quasars i n the L M T Strip (Epoch 2001.0) 90 Table X I I : Right Ascension Observing Window (1992) 91 Table X I I I : Spectrophotometric Survey: Predicted Night ly Performance . . . . 92 v i List of Figures Figure 1: Surface of a Rota t ing F l u i d 93 Figure 2: Effect of the Earth 's Curvature on the Surface of a Rota t ing F l u i d . 94 Figure 3: Effect of A x i s Misalignment on the Surface of a Rota t ing F l u i d . . . 95 Figure 4: East-West Response of C C D to a Del ta-Funct ion 96 Figure 5: East-West Response of C C D to a Gaussian 97 Figure 6: Star Tra i l Curvature 98 Figure 7: Definit ion of Star Tra i l Curvature Entrance Phase 99 Figure 8: North-South Response of C C D to a Gaussian 100 Figure 9: M i r r o r Ce l l Composi t ion 101 Figure 10: M i r r o r Ce l l Flexure: Variables 102 Figure 11: M i r r o r Ce l l Flexure: Coordinates 103 Figure 12: M i r r o r Ce l l Flexure: Boundary Condit ions 104 Figure 13: Longi tudinal Stabil i ty of the Mi r ro r Ce l l 105 Figure 14: T r i p o d Legs: Resonant Frequency 106 Figure 15: T r i p o d Legs: Moment of Inertia 107 v i i Acknowledgements It goes without saying that the one person to whom I am indebted the most is my supervisor, Pau l Hickson. After two years, I 'm st i l l constantly amazed by his enormous wealth of knowledge. His door has always been open, and his patience and support of me, despite my bone-headed blunders, has been remarkable. There are very few people (even fewer scientists), i f any, that I have met i n my life who I respect more than Paul ; there are none for whom I would rather work. I am excited about the opportunity to work as his P h . D . student for the next few years; my only goal is to improve during that t ime and contribute more meaningfully to our work together. Outside of Pau l , there are a few fellow scientists whom I would like to acknowledge. Each has contributed i n one way or another to my current enthusiasm for astronomy, whether it was introducing me to the subject, taking the t ime to acquaint themselves w i t h my work, or s imply providing an answer to some question. In no part icular order, my thanks go out to Richard E l l i s , D a v i d K o o , Scott Tremaine, Bruce Partridge, George Coyne, Jean Goad, Dona ld Groom, Leo B l i t z , M i k e F i ch , Andrew Potter, and of course our fellow collaborators, Ermanno Bor ra and Robert Content. Financial ly, none of my research could have been possible without the support of a Universi ty Graduate Fellowship from U B C . O n top of this, being awarded a 1990 Society for Photo-optical Instrumentation Engineering A w a r d was a great honour. M y thanks to bo th for providing the means for me to attend grad school. A s far as friends go, I've made a bunch, but there are three i n particular that I ' l l rave about: my fellow "Stooges", Moe " P i l l " Hodder (Thanks for Fortranning and S M i n g my Figures 5 and 8 Pi l l ! ) and Lar ry "Biffmeister" Kennelly, and Nadine " N D " , "Nad ia" , "Nadinsky" Dinshaw (for al l the secret reasons only we' l l ever know ... gig-gle). A l so on the friend front: Jaymie "Dr . L i b i d o " Matthews (should you find any aspect of this thesis coherent, it was probably due to his editing), D a n Hurley (who, if v m he doesn't get my Spinal Tap video back to me, will feel my wrath), Kalpana & "Gen-tleman" Jim Gilroy, Sally "Salamander" Craven, Dave "Unhhhhh" Woods, Claudia "Boogeda, Boogeda, Boogeda, Ciao, Click" Mendes de Oliveira, my fellow bald tongue-toucher "Little Jimmy" Brewer, Gordon "Gordinsky" Drukier, and Derek "Crazy Legs" Richardson (thanks for "providing" Cindy Crawford), Gerry "Big — Gerry" Grieve, Jill (sigh ... to drown in those eyes ...), Anne (my new $1.25 friend), Julia (despite her breathing problems), Karen, Sheila, Chris, ad infinitum. To Aii ... I'm sorry things had to end so badly, but you'll always be part of my memories. Special thanks go out to the Peace Arch Hospital Emergency Staff who saved my finger from an early death at the hands of a possessed table saw. Thanks to Maggie at Rehab Services for working me over after the broken shoulder. To a pretty cool family (Mom, Sissa, Gram & Gramp, Bev, Brian, Dana, Ian, Megan, Eric, Deb, Rachel, and Katy): see ya all at Christmas. With all that out of the way, let me acknowledge the truly important things in life: Lynyrd Skynyrd, Steve Morse, Sinead O'Connor, CSICOP, Star Trek, and the toughest little street-fighter ever, The Punisher (with apologies to Ronnie van Zant). Oh yeah ... and the best of them all, my little Kristine. ix Chapter 1 LIQUID MIRROR TELESCOPES 1.1 Introduction The arguments for the scientific gains to be made by the next generation of 10-meter class telescopes, especially as large-scale survey instruments i n observational cosmology, are well-documented (e.g. Hayakawa 1989; Kirshner 1987). W h i l e the last decade has seen remarkable advances i n detector technology, like the advent of highly efficient, large-format Charge-Coupled Devices, or C C D s (e.g. Janesick et al. 1989), the practical physical l imit to the diameter of conventional glass telescope mirrors, ~ 6 m , was reached over ten years ago (Schroeder 1987). The few 8- to 10-meter telescopes, which, using new materials and technology, w i l l be commissioned during the next decade, are expensive. The growing trend towards increasing the power of ground-based optical observations demands that newer, innovative mirror designs be entertained, i n order to complement these new telescopes. W i t h this i n mind, a collaboration between the University of Br i t i sh Columbia and Lava l Universi ty was begun i n late 1988 to explore one part icularly promising possibility, a L i q u i d M i r r o r Telescope ( L M T ) , which has a rotating, paraboloidal "pool" of l iquid metall ic mercury, as its highly-reflective pr imary mirror. B y necessity, the telescope is 1 a stationary transit instrument, observing a narrow strip of sky passing through the zenith. The lack of mechanical tracking is overcome, i n part, by employing a C C D at the mirror 's prime focus i n the T ime Delay and Integrate ( T D I ) mode (e.g. M c G r a w , Cawson, and Keane 1986; Section 2.2.3). The restriction to the zenith is an obvious drawback to many aspects of astrophys-ical research. Nevertheless, it is s t i l l very appropriate for many survey applications, including large-scale structure, galaxy evolution, deep Q S O and galaxy surveys, and the determination of cosmological parameters v i a the classical tests discussed by Sandage (1988). Therefore, the L M T design is directed toward those applications which need large samplings of data, not necessarily from any specific direction in space. These very same projects, requiring a great deal of dedicated large-telescope time, w i l l be the ones most difficult to accommodate in the very competitive schedules of the planned 10m instruments. Indeed, this is the prime motivation behind this L M T project: providing relatively inexpensive lOm-class telescopes for broad surveys that are impract ical wi th conventional instruments. The L M T offers several practical advantages over the construction and operation of conventional telescopes. Point ing, tracking, and telescope flexure problems are elim-inated. Zeni th observations minimise atmospheric extinction and refraction, and take op t imum advantage of local seeing conditions. Vary ing focal lengths can be accommo-dated by simply altering the mirror 's angular velocity (see Section 2.2.1). B u t by far, the most important advantage held by the L M T is its cost: an order of magnitude less for telescopes of the same aperture) (Borra 1982). The concept of an L M T has been extensively investigated over the last decade by E . F . B o r r a and his colleagues at the Optics Laboratory of Laval University (e.g. B o r r a et al. 1990, and references therein). Their research to date has culminated in a diffraction-limited f/2 1.5- meter diameter l iquid mirror (Borra et al. 1989; Bor ra 1988) whose image quality in laboratory tests is comparable to the finest quality glass or ceramic mirrors. The next logical step was to proceed wi th the construction of a 2 prototype L M T to be used in a more realistic observatory environment. This led to the collaboration between U B C and Laval , drawing upon the existing strengths of the two groups i n instrumentation and observational cosmology, to construct a 2.5-m class L M T to conduct astronomical surveys. Pre l iminary outlines of this project appear in Hickson et al. (1990), Henbest (1990), Gibson and Hickson (1990), and Gibson et al. (1989). Before discussing the physical principles behind an L M T , and some of its inherent l imitat ions, i n Chapter 2, the remainder of this chapter charts the history of the L M T concept from the 1850s to the present work. Chapter 3 provides a detailed description of the design and instrumentation of the U B C / L a v a l 2.7m L M T . Its anticipated per-formance and intended scientific applications, especially those related to observational cosmology, are outlined in Chapter 4. 1.2 History The historical development of the L M T as a useful astronomical tool can be traced through three distinct eras of progress since the 1 9 t h Century. These are discussed in the next three sections and summarised in Table I. A complementary treatment of the history of the L M T is provided by Steel (1984, 1986). 1.2.1 1850 - 1872 The first published reference to the use as an astronomical telescope of a parabolic mirror formed from a rotating vessel containing l iqu id mercury appears to be a letter from Ernesto Capocci of the Naples Observatory read before the Royal Academy of Be lg ium in 1850 (Mai l ly 1872). A subsequent search of Capocci 's personal memoirs and papers (S. Maffeo 1989, private communication) showed that he d id not follow up this concept. A similar (equally brief) allusion to this idea was given by a " M r . Buchan of 3 the U S A " , i n the late 1850s, as recounted by Sir Dav id Brewster i n his personal diary (Gordon 1870). W h i l e it is difficult to definitely identify who constructed that dist inction seems to belong to either Robert Carr ington or Henry Skey,' bo th Engl ish astronomers of note during the nineteenth century. W o o d (1909b) reports that Carr ington built an L M T driven by a steam engine, near Redhi l l , England as early as 1868, although a literature search of Carrington's numerous papers does not corroborate this claim. It is known that Carr ington used l iquid mercury at his observatory (Carr ington 1874), but i n conjunction wi th a conventional telescope to accurately measure the latitude of his site. The first published account of a working L M T was provided by Skey (1872), who, upon emigrating to New Zealand, constructed a ~0 .35m diameter telescope near Dunedin . To give the mercury-bearing container a rotation rate as uniform as pos-sible, Skey tr ied two different techniques, both of which yielded clear images of test objects i n the laboratory: (i) an electromagnetic engine whose velocity was regulated by a conical pendulum, and (ii) a small hydroelectric turbine, driven by a regular flow of water. B y varying the rate of flow, and hence the mercury's angular velocity, cu, Skey demonstrated the abil i ty to alter the focal length, / , of an L M T , v i a the well-known relationship between cu.and / (see equation (2.6)). 1.2.2 1909 - 1922 Skey's prel iminary work in the lab was followed by about 35 years of inact ivi ty in L M T research. It was not unt i l Prof. Robert W o o d of Johns Hopkins Universi ty began his well-documented series of experiments wi th l iqu id mirrors that interest was revived (Wood 1909a, 1909b, 1909c). Unl ike Skey, whose goal was simply to prove that an L M T could work, Wood in-tended from the outset to bui ld an instrument of optical quality suitable for astronomical research. To that end, his goal was to first quantify and then minimise the optically degrading effects caused by ripples i n the reflecting surface of the mercury mirror. 4 W o o d (1909a) started w i th a 0.18m mirror surrounded by an empty barrel to protect it from air currents. It was driven by a revolving ring of small magnets which "pulled" a concentric r ing of magnets attached to the mercury-bearing basin. In this manner, the mercury was insulated from any bumps that might be transmitted from the motor along the drive belt, a problem which had plagued Skey. In addit ion to vibrations and bumps from the dr iving mechanism, W o o d pinpointed three other major sources of ripples on the mercury's surface: (i) shaking due to the continual "grinding" of the bearing surfaces of the mercury basin itself, (ii) leveling the mirror inaccurately, resulting i n " t idal waves" in the mercury pool , and (iii) fluctuations i n the angular velocity of the basin. Th i s last problem proved to be the most troublesome to solve (Wood 1909a). Having identified the main ripple sources, W o o d then proceeded to construct a functional 0.51m L M T , situated away from the city's inherent vibrations, to minimise these unwanted mechanical effects. The mirror was set up on a concrete pad located at the base of a 14-ft (~4.3m) deep pit adjacent to his barn. The original p lan of applying power to the basin magnetically was abandoned, since the use of fine threads of India rubber for the drive belt, which transmitted li t t le or no vibrations, worked just as well, and was much simpler (Wood 1909b). The basin was then accurately leveled wi th a spirit level. The end result of this first attempt was the el imination of a l l mechanical sources of serious ripple except one: the angular velocity variations imparted by the 110V A C electric motor. B y obtaining photographic trails of stars, like 7 C y g , W o o d demonstrated another side effect of these velocity variations: a periodic change i n the mirror 's focal length over a ± 2 c m range (Wood 1909a; equation (2.6)). Despite these problems, Wood's telescope successfully resolved both the e 1 and e 2 Lyrae double star systems (with angular separations of 2".6 and 2".3, respectively). This work constitutes the first astronomical observations made wi th an L M T . Wood's final contribution to L M T technology was to investigate means by which the residual surface ripples could be eliminated, once mechanical design was optimised 5 (Wood 1909a). Wood concluded that the most practical method available was to cover the mercury surface with a thin layer of transparent glycerine or castor oil. This fluid resulted in complete damping of all surface waves within 1-3 wavelengths of their cre-ation, without any noticeable degradation in the optical quality of the image (Wood 1909c). Despite his early successes, Wood soon abandoned the LMT because he felt that its restriction to zenith observations made the astronomical applications too limited. Until 1982, the only other published reference to LMTs was a 1922 proposal by a "B.A. McA." to construct a 15.24m LMT in Chile (Rigge 1922)! The announced goal was to test the instrument in 1924 during the opposition of Mars. However, the incredible design flaws inherent in this proposal were exposed by Rigge (1922), and not surprisingly, this telescope was never more than a pipe dream. 1.2.3 1982 - Present The modern era of L M T research began with Ermanno Borra's landmark paper (Borra 1982). He reassessed the details of the theory, and practical limitations of LMTs as true astronomical tools, in the light of the technological advances since Wood's time. Many of the points raised in his paper will be addressed in Chapter 2. Borra proposed the use of near-frictionless air bearings upon which a mercury basin could be rotated by a synchronous motor driven by an oscillator-stabilized AC power supply. This would eliminate the various sources of image-degrading ripples in the mercury's surface, first recognized by Wood (1909). Borra and his colleagues at Laval University have spent the last decade refining the necessary basic technology that is necessary to implement a fully functional of LMT (Borra et al. 1984 - 1990). Some of their key results are summarised in Section 2.4. It should also be noted that independent of the research at Laval University, a team at Khar'kov State University was working on the construction of a lm //0.72 LMT (Vasil'ev 1985). Their design incorporated a modification to Borra's basic mirror structure similar to Wood's (1909a) suggestion, in which the mirror is placed in a 6 rotating system of intermediate damping liquids ( I D L ) . The concept is quite simple; a l iqu id ( in this case, water) is spun in some large container. W i t h i n this outer container, an inner floating vessel is placed which supports the rotating l iquid mercury mirror. In equil ibr ium, both the I D L and the vessel of mercury spin at the same angular velocity. Thei r results showed that these moving components are essentially self-centering and self-leveling. They report that the optical surface of their mirror is very close to the diffraction l imi t (although they provide no supporting evidence for this), as the surface instabilities are suppressed by the I D L system. The mirror 's surface quality was l imited only by external background noise (i.e., quality was independent of rotational velocity). Another benefit of an IDL- type L M T is that the entire base of the mirror is uniformly supported by the I D L , not just the part i n direct contact w i t h the air bearing. This removes any pract ical weight l imit on the size of the mirror. Despite the success of their laboratory prototype, no further reports on L M T re-search by the Khar 'kov State University group have been found i n the literature. Borra 's 1982 paper systematically re-explored each of major sources of ripple i n an L M T : (i) bearing/drive/floor vibrations, (ii) leveling inaccuracies, and (iii) rotational velocity fluctuations. The pr imary source of vibrations i n Wood's L M T was his inferior bearing mecha-nism. B o r r a (1982) has shown that the use of air bearings removes this problem and makes vibrations from the local environment the major source. Knife-edge tests by B o r r a et al. (1984) reveal the existence of low-amplitude, concentric ripples induced by vibrations from a nearby bui lding transmitted through the floor of the mirror lab. This is not expected to be a problem at an isolated mountain site. The surface waves that can arise from large leveling inaccuracies are easy to eliminate by using Wood's method of aligning the mirror 's rotational axis wi th a spirit level. The major problem for W o o d was drift i n the angular velocity, cv, of his mirror. Th i s velocity must be stable to better than one part i n a mi l l ion to maintain the required optical quality and a stable focal point (Borra 1982). This severe requirement is now 7 one of the easiest to solve, by dr iving the turntable wi th a synchronous motor (a luxury unavailable to W o o d at the turn of the century). Synchronous motors are ideal because they rotate at a speed controlled by the frequency of the voltage signal that feeds it (e.g. oscillator stabilized A C power supply). Quartz oscillators can provide a stable clock of accuracy better than one part i n 1 0 1 0 (Borra 1982), much greater than the necessary stabil i ty (i.e., 10 6 ) . Long-term stability (i.e., greater than one turn of the motor) of LO is assured by this motor, and the short-term stability (i.e., less than one turn) is controlled by inertia of the mirror rotating on the air bearing (Borra et al. 1984). Borra 's group at Laval University has investigated the feasibility of the L M T con-cept, i n a laboratory environment, over the past decade. The i r results provided the first quantitative analysis of the surface structure of a l iquid mercury mirror , including the stabili ty of focus, high spatial-frequency defects, and the mirror 's overall shape, each of which can vary w i th time (Borra et al. 1982-1990). Section 2.4 explores some of the results from Laval i n depth, the key points of which are summarised below. Detai led optical shop testing (e.g. Har tmann, Ronchi , knife-edge, direct imaging, and scatter-plate interferometric tests) has shown that large, diffraction-limited mirrors (e.g. f/2 1.5m) are now a reality (Borra et al. 1989; Beauchemin 1985). Pre l iminary research w i th a 1.2m / / 4 . 5 8 mirror i n the field, near Quebec Ci ty , showed that even wi th a rudimentary set-up (see Borra , Beauchemin, and Lalande 1985 for photo), that star trails of ~ 2" F W H M were attainable. The i r conclusion was that seeing at the sea level site was the l imi t ing factor for image attainment i n their system, not the l iquid mirror 's surface quality (Borra et al. 1988). These field results formed the basis of the first scientific paper based solely on the data collected from a L M T : a search for optical flares and flashes (Content et al. 1989). Recent results from the Laval L a b are discussed in Bor ra et al. (1990). 8 1.3 Summary Now that the success of the l iqu id mirror concept, first detailed i n the 1850s, has been so successfully exploited in the lab, the culmination of which being the realization of large diffraction-limited mirrors (Borra ei al. 1989), the time has come to explore the concept i n the field to better analyze its potential as an astronomical tool . The U B C / L a v a l collaboration was formed wi th this particular goal i n mind . The engineering specifics of this 2.7m L M T wi l l be discussed i n Chapter 3, while its scientific goals wi l l be addressed i n Chapter 4. First though, Chapter 2 introduces some of the general concepts and inherent complications that are anticipated. 9 Chapter 2 LIQUID MIRROR TELESCOPES: GENERAL CONCEPTS 2.1 Introduction The intrinsic characteristics of the U B C / L a v a l L M T , discussed i n Chapter 3, cannot be fully appreciated without an introduction to some of the underlying principles governing its operation and capabilities; this includes both the basic physics (such as forming the parabolic surface of the pr imary mirror) , and the logistics of obtaining useful astronom-ical data wi th a transit telescope. 2.2 Basics 2.2.1 The Surface of a Rotating Fluid The entire L M T concept relies upon one physical principle: that the equil ibr ium shape of the surface of a uniformly rotating fluid in a constant gravitational field is a paraboloid. 10 Consider a l iquid rotating about an axis of symmetry, say the z-axis, which is par-allel to the local gravitational field lines. A n y point P(r, z) on the surface experiences a centrifugal force, F c , and a gravitational force, Fg. U p o n reaching dynamical equi-l ib r ium, the components of these forces parallel to the tangent at P(r,z), F c | | and F 3 y must balance (Figure 1). Defining <f> as the angle between the normal to the surface and the gravitational force, Fg, we see that for |F 3 | | | = |FC|||, we must have \Fg\sm<f> = |F c |cos<£. (2.1) Thus t a n ^ = §4 = — , (2-2) substi tuting the standard expressions for the centrifugal and gravitat ional forces. F r o m Figure 1, we see that <j) is also equal to the angle between the tangent to the curve at the point P(r,z) and the r-axis. i.e., dz cj2r tan(/> = — = . (2.3) dr g Integrating (2.3) then gives 2 2 (2.4) 2<7 ' where the integration constant was set to zero, by placing the vertex of the paraboloid at the origin. In other words, equation (2.4) is a paraboloid of focal length / (Steel 1984), where r2 = Afz, 11 (2.5) and / = (2.6) 2.2.2 Liquid Mercury as a Mirror A n y l iqu id to be used as the surface of a telescope mirror must be highly reflective. The two candidates investigated to date are mercury (Hg) and gal l ium (Ga) . The latter is costly, and has an inherently high melting temperature (~ + 3 0 ° C for G a , versus ~ — 40° C for Hg; Dickson 1988), although eutectic gal l ium alloys may have lower melt ing temperatures ( E . F . B o r r a 1990, private communication). L i q u i d mercury has measured reflection coefficients of 79% at 3100A, 78% at 8700A, and 90% at 13000A, compared to freshly evaporated a luminum which has values of 92% (at 3200A) and 98% (at 10000A) (Borra 1982). However, the values quoted for a luminum do not consider the degradation i n reflectivity due to oxidation and dust accumulation on the surface. Whi le large mirrors are rarely re-aluminised more than once per year, l iquid mercury can easily be filtered as often as once a day, to continually mainta in its op t imum reflectivity of ~80% at visible wavelengths. Mercury is quite dense (~ 13.6 g /cm 3 ) , so to minimise the load on the air bearing, a th in layer is desirable. Therefore, the mirror cell core is usually buil t w i th an approx-imate parabolic cross-section to allow a uniform layer of mercury no thicker than about l - 4 m m (Bor ra et al. 1989). The evaporation of mercury at ambient temperatures does not pose a severe health problem, provided standard safety precautions are observed (Borra et al. 1985). A s w i l l be seen in Section 2.3.4, upon reaching dynamical equil ibrium, the mercury develops a th in transparent surface layer, which results in the near-elimination of harmful mercury vapours (Bor ra et al. 1990). One of the most important characteristics of any telescope mirror is its thermal coefficient of expansion, and this is also true for a mercury mirror. This is also relevant 12 to the contraction/expansion of the support t r ipod itself (a point which is addressed below, and again i n Section 3.3.3). It is easy to show that the change in the depth per m m , Ah, of a mercury layer, as the ambient temperature drops from TH to Ti, is given by the formula Ah = (2.7) where VH = £2t and VL — £2(t — Ah) are the volumes of rectangular boxes of mercury, each of mass l g , at the aforementioned higher and lower temperatures, respectively. For a box of depth, t — 1mm, and volume VH, £, the length of the remaining sides, is ~0.8592cm. A s a numerical example, consider a temperature change from + 2 0 ° C to + 1 0 ° C (typical during a night of observing). F rom the C R C Handbook of Chemistry and Physics, we know that VH and VL are then, 0.0738215 c m 3 and 0.0736877 c m 3 , respec-tively. Keeping £ constant (since the "boxes" are constrained i n the ^-plane and must expand/contract along the symmetry axis of the mirror) and using the values for VH and VL i n (2.7), leads to a contraction of ~ 1.8//m per m m of mercury for each 10°C decrease i n temperature. Contractions of this order are approximately three orders of magnitude smaller than the corresponding contraction of the instrumental support t r ipod (see Section 3.3.3), and w i l l be ignored i n the subsequent analysis. 2.2.3 Data Acquisition The main l imi ta t ion of an L M T is obviously the lack of mechanical tracking ability. It is restricted to observing the strip of sky which transits at the telescope's zenith, over the course of the year. It was this particular drawback which led W o o d (1909a) to dismiss the instrument as an oddity w i th l i t t le or no practical application. Only recently has detector technology advanced to the point where an L M T could make meaningful contributions to astronomy. The advent of electro-optical tracking wi th C C D s by groups 13 at Cambridge Universi ty (Wright and Mackay 1981) and the Universi ty of Ar i zona ( M c G r a w , Angel , and Sargent 1980) has ushered i n a new era for transit telescopes. The Cambridge technique is known as "drift scanning". Instead of using the C C D i n the standard mode (i.e., accumulating an image by tracking the telescope at the sidereal rate, then reading out the data after the exposure is complete), drift scanning "clocks" the C C D continuously at some given slower rate (e.g. one row per second), at the same time, mechanically moving the chip itself i n an east-west direction, to avoid image smear. In this way, the light from each object continues to accumulate on the same charge distr ibution. U p o n readout, each point in the sky has been sampled for an equal t ime by each pixel i n the column; hence, it has been detected wi th the mean efficiency of all the pixels i n the column. There wi l l be no detector nonuniformities along each column of output data (Mackay 1982). Nonuniformities between columns w i l l s t i l l exist, but these are easily and accurately corrected by a simple one-dimensional flat field. Conventional two-dimensional flat fielding can reduce C C D nonuniformities to 0.3-0.5% rms at best, whereas drift scanning can improve this to ^ 0 . 1 % rms (Mackay 1986). A s of 1986, this readout technique had been used to observe the faintest objects ever detected from the ground wi th I l l a - J magnitudes of ~ 2 7 m , using the 4m telescopes at Moun t Stromlo and K i t t Peak i n their "park" positions (Mackay and A s t i l l 1984; H a l l and Mackay 1984). Th i s improved threshold reflects the fact that the ultimate l imi t to the precision of C C D images arises from flat field nonuniformities ( M c G r a w et al. 1984). Bloemhof, Townes, and Vanderwyck (1986) have also exploited the drift scan technique wi th linear HgCdTe arrays to image very faint circumstellar dust shells, using the 3m I R T F on M a u n a Kea . Developed independently of the drift-scan approach is the "Time Delay and Inte-grate" (TDI ) technique of C C D readout (McGraw, Angel , and Sargent 1980; M c G r a w et al. 1982, 1984; M c G r a w , Cawson, and Keane 1986). Essentially, the T D I mode is a variant of drift scanning where the C C D is kept physically fixed, and rather than clock-ing at an arbitrary slow rate, the clocking is done exactly at the sidereal rate. The flat 14 fielding advantages of drift scanning apply equally. For what it loses i n exposure time per object per night compared to the drift scanning method (since integration time for T D I is governed by the physical size of the C C D ) , the T D I mode gains i n its simplicity of design and operation. Because of the digital nature of the images, observations from different nights can be co-added to increase the total integration time, i f necessary. The software and database support for the 1.8m C C D / T r a n s i t Instrument ( C T l ) of M c G r a w et al. has been extensively documented (Cawson, M c G r a w , and Keane 1986a, 1986b; Bernat 1986; Cawson and M c G r a w 1988). The T D I mode is currently i n use on the Palomar 5m telescope (again, i n the "park" position) as part of a grism C C D transit survey to search for quasars at large redshift (Schmidt, Schneider, and G u n n 1986, 1988). In fact, the most distant object known i n the universe (quasar P C 1158+4635, at a redshift of 4.73) was discovered as part of this T D I survey (Schneider, Schmidt, and G u n n 1989). 2.3 Complications Beyond the simple principles outlined above, there are numerous complications which must be handled i n detail to design a large L M T . The necessary equations are derived i n the next six sections, and are applied to the discussion of the U B C / L a v a l 2.7m L M T i n Chapter 3. 2.3.1 Curvature of the E a r t h A t the point P(r, z) shown in Figure 1, the gravitational field lines were assumed to lie parallel to the symmetry axis of the mirror. In reality, the Ear th 's curvature leads to the geometry shown i n Figure 2, where the gravitational field line at P(r, z) is inclined at an angle 6 to the symmetry axis. (The remaining angles were defined i n Section 2.2.1.) To achieve dynamical equil ibrium, the forces F c | | and F^jj must balance, so 15 | F 5 | c o s 0 s i n < £ = ( | F C | - |F 3 | s in0)cos(/>, (2.8) which leads to |F C [ - | F g | s i n ^ F c tan<2> = - 1 —' " ' = — — tan0. (2.9) r | F 9 | c o s 0 F 3 c o s 0 v ; Using equation (2.3), and the identity sec 9 = Vl + t a n 2 9, we can write (2.9) as dz uj2r dr g \ A + t a n 2 0 - t a n 6 > , (2.10) such that r = (r® + z ) t a n 0 , (2.11) where r® = 6.378 x 10 8 cm is the radius of the Ear th . Substi tut ing (2.11) into (2.10) (since z << r®) , using the Taylor series expansion for [1 — (T^ - ) 2 ] 1 / 2 , and integrating (such that the integration constant is zero), yields u2r2 r2 uj2r4 z — 1 2g 2r© 8#r | Equat ion (2.12) gives the resultant shape of the equipotential surface (which is identical to equation (2.4), if we neglect the Earth 's curvature; i.e., r® —> oo). The first-order corrective term in (2.12) is proportional to r 2 , which represents a simple shift i n focal length ( t r iv ia l to correct). The second- order term is proportional to r 4 , corresponding to the first pr imary Seidel wavefront aberration, spherical aberration (Welford 1986). We wi l l show later that the higher order terms are negligible. 2.3.2 Rotation of the Earth More important than the Earth 's curvature is the effect of Coriolis force due to the Ear th ' s rotation. Following Bor ra et al. (1985), consider a mirror rotat ing wi th angular 16 2.12 velocity u on the surface of the Ear th , which is rotating wi th an angular velocity Q = 7.27 x 1 0 - 5 rad/s, itself. The l iqu id experiences a Coriolis acceleration, a c 0 , given by a C o = 2 (wxr )xn = 2(w • ft)r - 2(r - n)w, (2.13) using the triple product rule. Thus, a c o = 2uQrcos ar — 2uClr cos /3k (2.14) a'c° «L C o where a is the mirror 's latitude, and is the azimuth of a given point on the mirror, r and k are unit vectors i n the radial direction and parallel to the symmetry axis of the mirror (i.e., the z-axis), respectively. The accelerations a £ 0 and a ^ 0 must be added vectorially to F c and Fg, respectively. So, equation (2.3) becomes dz | F C | + | 4J d r | F 3 | + | a g o l (2.15) i.e.. dz oj2r 2ujQ,r cos a 1 dr~ = ^~g~ + 9 1 -(2u>ftr cos f3)/g'' ( 2 ' 1 6 ) A Taylor series expansion for the second term i n parentheses i n (2.16) leads to dz u>2r 2uQrcosa 2u)3Qr2 cos B 4 u ; 2 f i 2 r 2 coscecos/9 + + = - + dr g g g2 g2 AuAQ?rz cos 2 /3 8 w 3 f i 3 r 3 cos a cos 2 j3 s ~ + IS - + ••• (2-17) 9 9 Integrating (2.17) to second order produces 17 z = + uQr2cosa 2u>3Qr3 cos /3 4u2Q,2r3 cos a cos /? 9 3fif2 3g2 u ; 4 f t 2 r 4 cos 2 /? 2 u ; 3 Q 3 r 4 cos a cos 2 /? (2.18) This is the resultant equipotential surface. (Again , neglecting the Coriolis effect (i.e., —> 0) recovers equation (2.4)). The first corrective term in (2.18) simply adds to the angular velocity ui, again, producing only a shift i n the focal length, / , which is easily correctable. The second and th i rd corrective terms are proportional to r 3 cos 0, corresponding to the second primary Seidel wavefront aberration, coma. The fourth and fifth corrective terms (proportional to ( r 2 ) ( r 2 cos 2 /3)) are a combination of the th i rd and fourth Seidel terms, astigmatism and field curvature (Welford 1986). 2 .3 .3 A x i s M i s a l i g n m e n t A n improper alignment of the pr imary mirror's symmetry axis w i th the local gravita-t ional field lines can generate "t idal waves" i f the misalignment is too large. Al though spirit levels allow alignment sufficient to eliminate this problem, small deviations can s t i l l introduce measurable optical effects. For the moment, let us assume that the Ea r th has no curvature. Now, we examine a situation i n which the symmetry/rota t ion axis z of the mirror is t i l ted at an angle 0 from the gravitational field lines, as shown schematically i n Figure 3. We have chosen the coordinate z to lie along the rotation axis (i.e., x and y i n the plane of the t i l ted mirror) . In this frame of reference, g and r are given by g = — g sin 0y + g cos 9z (2.19) and 18 r = r cos 8x + r sin /3y. (2.20) F rom Figure 3, by construction, we see that the centrifugal and gravitational forces, | F C | and | F 5 | , respectively, can be writ ten | F C | = u2r + g • f = u> 2rsin#sin/? (2.21) and | F 3 | = g cos 9. (2.22) Recal l ing (2.2) and (2.3), we know dz dr | F , | so dz "2 dr g cos 9 Integrating equation (2.24) then gives u2r2 (2.23) U r - tan9sin/?. (2.24) z = sec 9 — tan 9 sin /?r, (2.25) which, through Taylor series expansions for sec 9 and tanf5, gives the expression for the effect of axis misalignment on the parabola's figure: oj2r2 . . n . „ .u2r2.nr, .sinBr. z = —-(sm8r)9 + (-I-)92-(—^)93 + .... (2.26) The first corrective term signifies distortion (Welford 1986), while the second-order term corresponds to astigmatism. 19 2.3.4 Surface Waves O f pr imary importance in the design of any L M T is avoiding waves on the surface of the mercury which w i l l degrade the image. This section introduces a very simpli-fied quantitative analysis of a much more complex problem. The reader is directed to Bor ra , Beauchemin, and Lalande (1985) for their discussion of the analytical vs. numer-ical aspects of the surface wave problem. The analysis below follows the terminology and format given i n Landau and Lifshitz (1959). Detai led physical descriptions of the equations used here can be found in that reference. Bor ra et al. (1985) found that mirror vibrations generate concentric waves at the mercury's surface. The angular frequency of these waves, u>, defined by u = 2xu, (2.27) is described by the dispersion relation OJ2 = (gk + — ) t a n h ( M ) , (2.28) where the gravitational acceleration, g, is ~981 c m / s 2 ; h is the mercury layer depth; p is the mercury's density (~13.57 g / c m 3 ) ; a, the mercury's surface tension, is ~487 dyne/cm; and k is the wave number for a wavelength, A, defined by k = y . (2.29) Also important are the wave packet's group velocity, vg (i.e., its physical propagation velocity), and its phase velocity, vp, given respectively by "» = Tv (2'30) and, 20 (2.31) We consider the two extreme wavelength regimes, long (i.e., gravity waves) and short (i.e., capillary waves). Gravi ty waves arise from the force of gravity acting upon the surface of the mercury, while capillary waves, or ripples, result from the action of surface tension. Borra , Beauchemin, and Lalande (1985) have discussed surface wave theory only i n the context of generation by wind, not by vibration. Thei r results suggest that capillary, self-induced waves of A < h (i.e., not externally generated by wind, which can be eliminated by a silo structure and proper baffling) should be the only serious concern for l iqu id mirrors, even though they have yet to be observed (Bor ra ei al. 1988). The dissipation rate for wind waves is proportional to k2 (Borra , Beauchemin, and Lalande 1985), and increases rapidly for wavelengths less than 1cm. For deep-water capillary waves shorter than 0.5cm, dissipation dominates and the l iqu id should be stable at laminar windspeeds below ~6.1m/s . Th i s is beyond the r i m speed of most feasible L M T s , including the U B C / L a v a l L M T (~1.3m/s) . The earlier results of B o r r a et al. (1985) appear to confirm this prediction. Bor ra , Beauchemin, and Lalande (1985) also state that to maximize damping of these wind waves, the mercury layer should be as th in as possible. Corotat ion of the air near the mercury's surface could also be introduced to minimise the effects of wind. Explor ing the effects of self- and naturally-induced wind waves is one facet to be studied wi th the U B C / L a v a l prototype telescope. Because the generation of vibration-induced waves is easier to quantify and treat analytically (Borra , Beauchemin, and Lalande 1985), we restrict the remaining discus-sion to this problem. In the long wavelength l imi t , kh « 1 (i.e., v « Vp/2-Kh, from (2.29) and (2.31)), equation (2.28) can be writ ten 2 ghk2. (2.32) 21 B y comparing (2.30), (2.31), and (2.32), we see vg = vp = y/gh, (2.33) which means that, i n the long wavelength l imi t , " « ( 2 ' 3 4 ) From Landau and Lifshi tz (1959), the characteristic damping time for gravity waves, TG, is given by g2p 2.6 x 10 5 . T ' = ^ K ( 2 ' 3 5 ) where rj ~0.01615 g / s / c m is the viscosity of mercury. Equat ion (2.35) implies that i n the frequency regime defined by (2.34), the gravity-wave damping t ime is very large. This is easy to satisfy by ensuring that the fundamental resonant frequency of the mirror cell (see Section is ^ ^Jg/h/2ir. Numerical examples w i l l be provided i n Section 3.2.2. In the short wavelength l imi t , kh » 1 (i.e., v » vp/2Trh, from (2.29) and (2.31)), equation (2.28) becomes 2 afc3 OJZ « , (2.36) and the phase velocity is 2 2ira v* = r> = V T T ' (2-37) so, the sort-wavelength l imit is set by 22 The characteristic damping time for capillary waves, r c (Landau and Lifshitz 1959), is given by o ^ V 3 394.25 , s imply ing that capillary waves vanish quickly if v is sufficiently high. Determining the amplitude, A, of the unavoidable capillary waves is an important aspect in the study of surface phenomena, but a difficult one theoretically (again, see Bor ra , Beauchemin, and Lalande 1985). However, we can make a few qualitative state-ments regarding capillary wave amplitudes, which wi l l resurface i n Section 3.2.2. The mechanical energy dissipation of a surface capillary wave, \ E \ , is related to its average energy, E , by the relation (Landau and Lifshitz 1959, p.99): \ E \ = — , (2.40) where r c is given by equation (2.39). Now, from Hall iday and Resnick (1978, p.416), we also know that \ E \ oc A2v2. (2.41) Combining (2.40) and (2.41) yields A cx — L - V S . (2.42) \/TCV Substi tut ing (2.39) i n (2.42) then provides a proportionality between the frequency of a ripple and its amplitude: A oc - \ J - \ [ E . (2.43) 23 This is another compelling reason to set the resonant frequency of the loaded mirror as high as possible; the higher the fundamental frequency of the mirror u, the smaller the amplitude of any resultant capillary waves (see also Chapter 3). Obviously, decreasing the damping times of induced surface waves is a major goal i n the mirror design. A promising method first explored by W o o d (1909a), and currently under investigation by Bor ra et al. (1985), is the use of a th in , damping layer of l iquid (e.g. a transparent viscous fluid like glycerin) on the surface of the mercury itself. Wha t effect would such a th in film have on the damping time of capillary waves? The damping t ime of the th in f i lm, 77 (Landau and Lifshitz 1959), is given by 2 5 / 2 a l / 3 1/6 The ratio of this to the damping time for capillary waves (equation (2.39)) is (2.45) or, using (2.27), IL = 2 s / 3 ( ^ ) i / 6 « n . 08^ 1 / 6 . (2.46) Thus, the presence of a th in film on the mercury's surface" leads to large decrease i n the capillary wave damping time (in the frequency range under discussion here, given by (2.38)). A s a final note, the stabilization time for the mercury surface of the 1.5m l iquid mirror (Borra et al. 1989) following a "cold" start-up is on the order of three hours (Borra et al. 1990). A chemical reaction occurs between the mercury and the polyester resin used on the surface of the mirror cell which creates a transparent surface layer. This promotes the stabilization process, and also helps damp any surface waves (Borra et al. 1990). The chemistry of this reaction is st i l l being investigated ( E . F . B o r r a 1989, private communication). 24 2.3.5 TDI CCD Readout Mode A s first noted by M c G r a w , Angel , and Sargent (1980), images acquired using the T D I mode have an elongated shape i n the east-west direction, because the rows must be shifted discretely, whereas the incoming image moves across the C C D continuously. The resulting image is composed of the convolution of the seeing disk wi th the C C D pixel wid th . This section presents the first published quantitative description of this elongation effect. M u c h of the mathematical foundation for this work can be found in Bracewell (1978). The response function for a C C D i n the east-west direction is generated by simply passing a delta-function, 6(9), along a single column of the detector. For a particular pixel , n, and offset LO from 6(9) to the center of some reference pixel (say pixel n = 0, for which u> could range from -1/2 to +1/2 pixel) when the C C D clocking occurs, the response can be uniquely determined by simply shifting a triangle function, A(9) (see Figure 5b) appropriately and sampling the function at the centers of the pixels underneath A(9). Figure 4 illustrates the procedure. For a part icular n and u>, the ordinate of Figure 5, 9, is given by to be and therefore the intensity of a given pixel , 1(6), is found by substituting n and u> into (2.47) to determine 6, and then using Figure 5b to derive 1(9). Represented mathemat-ically, the resulting intensity is the convolution of 6(9) wi th A(9). i.e., K n o w i n g the response function to an input delta-function, the next step is to de-termine 1(9) for any general input function, f(9). Aga in , mathematically the problem reduces to evaluating the convolution of f(0) w i th A(9). i.e., 9 = n — u> , 2 (2.47) (2.48) 25 /oo f(y)A(9 - y) dy. (2.49) -oo F r o m Figure 5b, we can see that A(9 — y) = 0 when (a) 9 — y = — 1 (i.e., y = 9 + 1), and (b) 9 — y = +1 (i.e., y = 9 — 1). Outside these extrema, equation (2.49) reduces to zero. Incorporating these new integration l imits , (2.49) becomes m = f 1f(y)A(9-y)dy. (2.50) rd-l The point at which the two linear portions of A(9 — y) meet (i.e., 9 — y = 0) is y = 9. Integral (2.50) can then be split i n the following manner: 1(0) = f f(y)A(9-y)dy + f 1 f(y)A(9-y)dy. (2.51) Je+i Je In the range 9 + 1 > y > 9, we see from Figure 5b that A(9-y) = 9-y + l. (2.52) Similarly, i n the range 9 > y > 9 — 1, we have A ( 0 - y ) = - 0 + y + l . (2.53) Substi tut ing (2.52) and (2.53) into the appropriate integrals of (2.51), the desired rela-t ion for the response to a function f(9) is given by 1(9) = (1 + 9) f f(y)dy - f yf(y)dy + Je+i Je+i re-i r-e-i (1-9) f(y)dy + / ' yf(y)dy. (2.54) Jo Je Let us now explore the effect of this convolution upon a stellar point-spread-function ( P S F ) , chosen to be a simple Gaussian of the form 26 fW = e X p - eo)2/2a% (2.55) ay lit where a is the standard deviation, and 90 is the offset of the Gaussian's peak from the center of the reference pixel at the time when the C C D clocking occurs (taken to be zero i n the calculations which follow). A Gaussian w i th 90 = 0 is shown in Figure 5a. The Gaussian has been normalised such that the area under the curve remains equal to unity. Its max imum is equal to Xjas/^K « 0.3989/cr, and the F W H M is given by \J2 In 2cr « 2.355<r. Substi tut ing (2.55) into (2.54) leads to 1(9) ^~7= I exp[-y2/2a2]dy )= f y exp [-y2/2a2] dy + ay/Air Jo+i a\J In Jg+i (I — Q) \ rtt-1 exp[-y2/2a2]dy + —== yexp[-y2/2a2]dy. (2.56) crv27r Je a\/2ir Jo rO-l i O-l i.e. 1(9) = ^ 3 f exp[-y2/2a2]dy + ^ - = 2 f * exp [-y2/2a2] dy + - £ = [ 2 exp \-92/2a2\ - exp [-(9 + l)2/2a2} - exp [-(9 - I)2/2a2]]. (2.57) V 2TT One numerical application of this convolution calculation is now given. A n in i t ia l Gaussian wi th a standard deviation a = 1.370 pixels (corresponding to a Ford 2048 x 2048 C C D ' s 0".62 pixels operating under 2".0 F W H M seeing), was used i n (2.57). Recal l ing that 90 = 0 in this example, the convolved output, 1(9), found by numerically solving (2.57), is compared wi th the input Gaussian, f(9), in Figure 5c. The F W H M 27 of the convolved output is ~ 2".09, as opposed to the input F W H M of 2".0, while the central peak is approximately 95.8% of its in i t ia l value. It is important to remember that this small elongation of the F W H M of the seeing disk is only seen i n the east-west direction of the images. Its effect on automatic object classification by shape parameters alone is s t i l l under investigation. Vary ing 0O s imply results i n a linear shift of the convolved output, 1(0), along the ordinate-axis of Figure 5c, i n keeping wi th the resultant shift of f(0). A s the offset of the Gaussian peak, 0O, is always unknown, it is difficult to de-convolve the output profile to obtain the "true" input profile, and its proper centroid location. Fortunately, for many of the surveys intended for the L M T , shape parameters are not the key factor i n object classification. For example, spectral energy distributions (SEDs) of a survey object (see Chapter 4) can be matched to reference S E D s of various galaxy types, stars, etc., to classify it . 2.3.6 Star Trail Curvature A second image distortion is caused by star t ra i l curvature occurring at non-zero lati-tudes, which leads to an asymmetric north-south elongation, quite independent of the east-west effect discussed above. Figure 6 schematically shows a typical star t ra i l across a C C D , as observed from the zenith, Z. W h a t is the the max imum distance, A , that a star deviates from the center of a given C C D column? Let a be a segment of the great circle connecting the nor th celestial pole, N C P , to the zenith, while b is the port ion of an orthogonal great circle perpendicular to a (found by projecting the C C D onto the celestial sphere). B y definition, therefore, ZD is 90° . The spherical cosine law gives cose? = cos a cos b + sin a sin b cos D. (2.58) B u t for D = 90° , 28 cose? = cos a cos fe. (2.59) B y construction from Figure 6, the maximum deviation, A , is given by A = d — a, so cos(A + a) = cos a cos fe, (2.60) which expands into 1 —tan A tan a = cos b sec A . (2.61) Since A is small , we can use Taylor series expansion for tan A and sec A to first order approximation ( A and 1, respectively) i n (2.61) to find A = (1 - cos fe) cot a, (2.62) where a is the telescope's colatitude (i.e., a = 90° — £, where £ is the latitude), so finally A = (1 - cos 6) tan £. (2.63) Therefore, the telescope's latitude, I, and half the detector's angular field of view, fe, specify the max imum deviation, A , of a star t ra i l from the center of a C C D column. For the U B C / L a v a l L M T , situated near Vancouver (£ ~ + 4 9 ° . 0 6 ) , a Ford 2048 x 2048 C C D is intended as the imaging device. The field of view of the detector w i l l be 21'07" x 21'07" (i.e., 6 ~ 10'03".5 = 0 M 6 7 6 ) . Using (2.63), we expect that for our instrument, a t ra i l w i l l deviate by as much as A=1.64 pixels, the result being an image elongation i n the north-south direction. We quantify this elongation i n the same way as the east-west effect, i n Section 2.3.5. F i rs t , we numerically determine the response function, h(6), along the north-south axis of the C C D . A s before, we consider the response to a delta-function, which this time 29 follows the path of a star/galaxy t ra i l , as a function of the "entrance phase", to, where 10 is defined i n Figure 7. The result is presented i n Figure 8b, where the ordinate, 0, is, 0 = n — oo; n refers to a C C D column (see Figure 7). The response function, h{9), can now be convolved wi th any general input function, / ( 0 ) , as before, to determine the effect of t ra i l curvature on the north-south structure of images, i.e., /oo f(y)h(9-y)dy. (2.64) -oo From Figure 8b, we know that h(9 — y) = 0 for 9 — y < — 1 and 9 — y > +2, so the new integration l imits lead to m = f 2 f(y)H6-y)dy, (2.65) Je+i which can subsequently be split at 9 — y = 0 to yield 1(0)= f f(y)h1(9-y)dy+ f 2 f(y)h2{9-y)dy, (2.66) Je+i Je l(9-y)dy+  2 where h\(9) covers the range — 1 < 9 < 0, and h,2(0) covers 0 < 9 < 2 (Figure 8b). Least squares fitting of fifth-order polynomials to each of these ranges of Figure 8b, gives the following approximate analytical expressions for hi(9 — y) and h2{9 — y): h(9 - y) w 0.74374 + 0.62067(0 -y) + 2.4719(0 - yf + 8.8109(0 - y)3 + 12.023(0 - yf + 5.8045(0 - y ) 5 , (2.67) h2(0 ~ y) « 0.72856 - 1.9903(0 - y) + 4.6647(0 - yf - 5.3701(0 - y ) 3 + 30 2.7456(6 - yf - 0.51668(0 - yf. (2.68) Note that, as i n Section 2.3.5, where J A(9)d9 — 1, we also have j h(6)d6 = 1. Now, using (2.66), (2.67), and (2.68), i n conjunction wi th an input function, f(6), allows us to determine the output image intensity, 1(9). In Figure 8c, the convolved output (found by numerically solving (2.66)) is shown along wi th the input Gaussian of equation (2.55). Note that the convolved image is slightly asymmetrical . The F W H M of the output is ~ 2". 18, as opposed to the input F W H M of 2".0, while the maximum has been reduced to ~90.7% of its in i t ia l value. The peak of the convolved output has also been shifted by a negligible amount of only ~ 0".04 to the right. 2.4 The Laval Experiments The in i t ia l experiments of the Laval group are documented in three references: Bor ra ei al. (1984,1985); Beauchemin (1985). The three main concerns of this early work was to better quantify the stability of focus, high spatial-frequency defects, and the overall shape of the mirror , a l l of which can vary rapidly w i t h time. T w o mirrors were used predominantly, a 1-meter f/1.6 and a 1.65-meter / / 0 . 8 9 . The first two concerns just l isted were addressed by the knife-edge test (Ojeda-Castaneda 1978), and supplemented by direct imagery of a standard resolution bar-chart. A Har tmann test (Ghozei l 1978) was employed to estimate the overall shape of the mirror. Each of these tests was performed at the mirror 's center of curvature, hence spherical aberration was a problem, and i n fact the mirror had to be diaphragmed to 0.4m for some of the tests, because the group d id not have access to corrector lenses for this early work. The mirrors themselves had a very simple composition, a stiff p lywood base wi th a metal retaining strip around the outside. The final surface was a spuncast polyester resin. The mercury layer then acts as a l iquid high-reflectivity coating of four to seven 31 millimeters i n thickness. Surface tension tended to break the mercury into puddles if thinner layers were used (Borra et al. 1985). B o r r a et,al. (1984,1985) used a modified linear Har tmann test (Ghozei l 1978), which incorporated a screen which had 18 holes, 12.7mm in diameter, i n a single row, w i th 50.8mm separation between each of the holes. They would sample one direction at a t ime, then position to a different angle, etc., so as to detect defects such as standing waves, wi th respect to the laboratory frame. Exposure times of ~100 seconds were used (i.e., seeing effects were not important) . Deviations from a perfect paraboloid having an rms amplitude of about A/10 at 5000A were seen. A problem w i t h this optical test is that one cannot study surface quality on scales smaller than the linear space between the screen's holes (i.e., 50.8mm), a point which w i l l be referred to shortly. The second optical test employed was the knife-edge test (Ojeda-Castaneda 1978), a test renowned for its extreme sensitivity. In fact, it can detect defects such as small as A/600, as well as time-varying defects such as traveling waves or a t ime-varying focal length. T w o defects were found at this stage: (i) the surface of the mirror "puckered" around dirt particles, due to surface tension, that had accumulated on the mirror, resulting i n the introduction of scattered light, and (ii) concentric rings were seen on the surface, which were apparently caused by vibrations communicated through the floor, originating i n a nearby building, not from the drive/bearing system. The first defect is easy to control as mercury can be skimmed/fil tered on a daily basis, i f so desired. The second problem can also be minimised by constructing a support mount which contains a large amount of damping and possesses a resonant frequency as far as possible from the peak of the vibrat ion spectrum, as it is background noise which excites the resonant frequency of the three-point mount on which the mirror rests (Borra et aZ.1985). Beauchemin (1985) showed that the amplitude of these concentric ripples was i n the range A/10 to A /15 , at 5000A. The knife-edge test was supplemented by directly imaging a resolution bar-chart. Like the knife-edge test, this imaging reveals high spatial-frequency defects which are 32 not detected by the Har tmann test. The Bor ra et al. (1984,1985) papers showed that a resolution of ~ 0".75 had been realized, very close to the theoretical resolution expected from this diaphragmed l m mirror. A faint fuzz seen around the bars i n the chart corresponded to a ripple amplitude of ~ A/10 , i n agreement w i th the knife-edge tests (Beauchemin 1985). The above tests were replicated after introducing a thin layer of glycerin onto the mercury's surface (Borra et al. 1985). Glycer in does not affect the system's reflectivity, and has the advantage of being very insensitive to vibrations and wind-induced surface disturbances. In fact, no ripples were detected wi th the knife-edge test. Unfortunately, the viscosity of glycerin d id pose a problem as nonhomogeneities which are generated while pouring the l iquid were s t i l l present several hours afterwards, whereas the mercury surface without the glycerin normally takes on the order of ten minutes to stabilize (Bor ra et al. 1985). These variations i n the glycerin thickness led to a substantially poorer image quality. T w o final points addressed i n this early work are: (i) mercury tarnishes slowly (on the order of one week before there is a noticeable decrease i n reflectivity), but can be cleaned on a daily basis, i f so desired, and (ii) damping times are short; disturbances having amplitudes greater then 1mm dampen out i n a few seconds (Bor ra et al. 1985). The second stage of the Laval group's work was to apply the success of their labo-ratory mirrors to a working L M T i n the field. T w o mirrors, one a l m / / 4 . 7 , the other a 1.2m / / 0 . 8 9 , were operated over two consecutive summers on the campus of Laval University, i n Quebec Ci ty . The pr imary references include: Bor ra , Beauchemin, and Lalande (1985); B o r r a et al. (1988). The first summer, the l m f /A.l mirror was used exclusively. Longer focal lengths were used i n the field than were used in the lab because they d id not have access to any corrector lenses. C o m a would have made the usable field of the l m f/1.6 mirror too small . The support for the mirror was very simple; there was no bracing, or foundations. 33 A photograph of the rudimentary setup can be found i n Borra , Beauchemin, and Lalande (1985). The pr imary goal of this in i t ia l observing season was to acquire data for star trails as they pass through the zenith, using a 35mm photographic camera wi th its shutter left open. Focussing errors i n ths crude setup dominated, and errors as large as 1" were reported. S t i l l , the F W H M of the star trails was ~ 2"; an excellent result considering the site's sea level location on the Laval Universi ty campus, as well as the accompanying focus errors just mentioned. The star trails wiggled w i th an apparent amplitude of ~ 1" at a t ime scale of ~ 2 seconds. Seeing induced image motion due to the location near a warm bui lding is the most probable cause of this effect (Borra , Beauchemin, and Lalande 1985), although some inherent structural movement of the mirror 's frame cannot be excluded. The one detectable problem wi th this original data was an accompanying faint "fuzz" surrounding the trails due to the same type of low-amplitude ripples that were seen during the optical shop tests (Borra et al. 1984). The estimated fraction of light contained i n the central peak, found by scanning across the star t ra i l itself, was ~70%, w i th the remaining being found i n the rings. This corresponds to surface deviations having amplitudes of ~ A/16 for sinusoidal ripples (Beauchemin 1985). Bor ra , Beauchemin, and Lalande (1985) also" found that the mercury surface could not tolerate turbulent winds much stronger than 10 k m / h r , although no thorough study was done. A silo-structure was suggested to minimise this problem, and has been incorporated into the U B C / L a v a l 2.7m L M T . The 1.2m / / 4 . 5 8 mirror was the pr imary instrument for the second observing season (Bor ra et al. 1988). Similar conclusions were found (~ 2" F W H M ) from the accumu-lat ion of more than 200 hours of data. The main difference in the second season's observing procedure was the introduction of a mylar cover to shelter the mirror from the wind. The result of this step was the unavoidable added presence of some scattered 34 light. For a site wi th 2" to 3" seeing, the mylar does not affect the images noticeably, but at the l " - leve l , there may be some problems. Monomolecular layers of o i l which are effective at dampening wind-induced waves were also utilised and w i l l be involved wi th future experimentation. The results of this second season formed the basis of a scientific paper whose goal was to put better constraints on the occurrence of optical flares and flashes (Content et al. 1989). The last two years of research at Laval has involved further detailed optical shop testing of a 1.5m f/2 mirror (Borra , Content, and Bo i ly 1988; B o r r a et al. 1989). The Har tmann test used by Bor ra et al. (1985) has been dropped i n favour of scatter-plate interferometric techniques (Maffick 1978). The Ronchi test (Cornejo-Rodriguez 1978) was also used by Bor ra , Content, and B o i l y (1988). N u l l Offner lenses have also been added to correct for the large spherical aberration that occurs at the center of curvature of the f/2 paraboloid, thus el iminating the need for diaphragming the mirror to an appropriate diameter (as was done by Bor ra et al. 1985). A resolution of ~ 0".4 wi th a five second exposure was seen. The image quality was inferior in one direction. This was suggested to be due to aberrations introduced by the nul l lenses (most likely a misalignment). The Ronchi test shows that image quality seems to be l imi ted by seeing i n the testing tower, as well as by print-through effects from the surface of the container (Borra , Content, and Bo i ly 1988). Print- through arises because the l iqu id flows slowly across the surface of the container. These low-speed flows are caused by alignment uncertainties, as well as from the friction of the air on the surface of the l iquid . B y resurfacing the container wi th an epoxy resin, the print-through is expected to decrease below the detectable level (Borra , Content, and Boi ly 1988; B o r r a et al. 1989). A recent technological development in the lab has been the abil i ty to establish thinner mercury layers on the mirror cell (Borra , Content, and B o i l y 1988). In the past (Borra et al. 1984), layers could not be less than 4mm, or else surface tension 35 would break the mercury into puddles. Currently, channels l c m wide and 0.5cm deep are formed around the periphery of the cell. The interaction by surface tension wi th the container now only takes place at the edges where the layer is deep, allowing a very th in layer everywhere else. T h i n layers are much less sensitive to vibrations and wind, because waves interact wi th the container's bot tom, thus increasing friction (Borra, Content, and Bo i ly 1988). A s such, it was no longer necessary to shelter the mercury from the air wi th mylar. A 2.5mm layer was used to establish the results reported here. A s the dominant cost of their instrument was the air bearing itself, reducing the thickness of the dense mercury layer is obviously of prime importance, as it reduces the weight that the bearing must support. Recently, B o r r a et al. (1989) have observed the actual A i r y disc of their 1.5m f Ji mirror. Diffraction rings are seen at the predicted locations, although the intensity and symmetry of the rings varies, possibly due to seeing i n the testing tower. This is the first time that the diffraction pattern of a large mirror has been directly observed, despite imaging through two nul l lenses, two folding-flat mirrors, and a beam splitter. Imperfections and alignment errors of these additional optical elements are expected to introduce some aberration. Interferometric measurements confirm that the mirror is indeed diffraction-limited. The central part of the A i r y pattern is produced mostly by low spatial frequency components of the mirror 's surface, whereas high spatial frequency ripples are respon-sible for extended "wings" and scattered light far from the center of the P S F . A s before (Bor ra et al. 1985), low amplitude high frequency ripples originating from the local environment are observed. The peak-to-valley amplitude of these ripples is less than A/30 , and contain only 3% of the total energy of the A i r y pattern (Borra et al. 1989). The P S F s do not show any detectable energy beyond 20" from the P S F center, thus these vibrat ion effects should be unimportant, especially when the observatory is moved to an isolated site. 36 The mirror surface is accurate to A/17 rms deviation, wi th a corresponding Strehl ratio of ~0.76 (Borra et a/.1989). Note that a surface accurate to better than A/14, or a Strehl ratio >0.8 can be considered to be diffraction-limited (Schroeder 1987). In other words, diffraction itself, rather than the aberrations of the system, is the main contributor to image degradation. So, this 1.5m f/2 l iqu id mirror can be considered essentially diffraction-limited. A summary of recent activities of the Lava l group can be found i n B o r r a et al. (1990). 2.5 Summary T w o sources of image degradation which cannot be totally eliminated are: the T D I readout technique, itself, and the curvature of star/galaxy trails across the detector at non-equatorial latitudes. The T D I technique i n the U B C / L a v a l L M T unavoidably "spreads" the F W H M of the intrinsic "east-west" P S F of a stellar-like object by ~ 5 % , while reducing its peak intensity by ~ 4 % . Similarly, for our instrument, star t ra i l curvature at the lati tude of the observatory causes the "north-south" P S F to "spread" by ~ 9 % , and drop its peak intensity by ~ 9 % . These image-degrading effect w i l l not adversely affect our observing program, which does not rely on fundamental shape parameters, but could lead to problems for other programs. Further numerical analysis of the effect is s t i l l underway. 37 Chapter 3 UBC/LAVAL 2.7m LMT: DESIGN AND STRUCTURAL ANALYSIS 3.1 Introduction Now that Chapter 2 has introduced many of the fundamental principles governing the operation of an L M T , we can now apply many of these concepts specifically to the U B C / L a v a l 2.7m mirror. Engineering aspects of the mirror, support t r ipod, and prime focus instrumentation package are addressed i n the subsequent three sections. 3.2 Mirror 3.2.1 Mirror Cell One of the unique design aspects of the 2.7m telescope is its Kevlar-covered foam mirror cell. Its composition, structure, and flexure characteristics are described below. Details of the stress analysis can be found in Appendix A . 38 Composition The pr imary considerations guiding the design of the mirror cell are supporting the mercury w i th min imal flexure and vibrat ion. Given the high cost and l imited load-bearing capacity of air bearings, it is imperative that the mirror cell material have a high strength-to-weight ratio. It was decided to abandon the p lywood base and metal retaining strip used i n earlier mirrors (Borra et al. 1984, 1985) and employ composite laminates, such as those popular in the aerospace industry. The core of the cell is Dow styrofoam wi th a density of 2 £ b / f t 3 (pj ~0.032 g / c m 3 ) . Th i s material was chosen for its good stability under loading, as well as ease of shaping by cutt ing wi th a hot-wire. A rough parabolic figure was cut into the surface of sixteen foam segments using this procedure. The segments were then glued together around a central a luminum hub. The laminat ing material (i.e., skin) used to cover the foam core is bidirectional weave Kev la r fabric (two layers of density ~1.39 g / c m 3 ) in an epoxy resin/glass microsphere matr ix . Kev la r was selected over fiberglass due to its superior damping characteristics and higher tensile modulus of elasticity (Es ~ 3.1 x 10 9 N / m 2 ) . U p o n application of the Kevla r skin, the parabolic surface was found to be accurate to ^ 5mm. A final ~5mm layer of spuncast, pure epoxy-resin gives the basic mirror cell a parabolic accuracy ^ 0.1mm over the entire surface. Channels 1cm wide and 0.5cm deep were dug in this epoxy surface around the inside and outside edges to l imi t surface tension between the mercury and cell to the periphery of the container. This allows one to deposit thinner mercury layers on the cell (Borra , Content, and Bo i ly 1988). Our 2.7m mirror w i l l operate wi th a reflective coating of mercury ~ 2 m m thick. Details of the mirror 's mercury surface w i l l be discussed i n Section 3.2.2. Figure 9 provides a cross-sectional view of the 2.7m mirror cell, as well as an expanded view of the skin and mercury coating. 39 Flexure One can treat the mercury layer as a th in reflective coating, much like a luminum on a glass mirror. A s in any telescope design, flexure of the mirror cell is one of the most cri t ical aspects of the L M T . This section presents a basic analytical solution to the flexure problem, and lays out the framework for a more rigorous numerical solution. The terminology and approach adopted here follows that of Landau and Lifshitz (1975, 1986). In cyl indrical coordinates, the equil ibrium equations governing the stress and strain of the flexed mirror cell, are given by equations (A.28) and A.36) , both derived i n Append ix A . i.e., Id, . darz and I d . . dazz Before we can solve the equil ibr ium equations for flexure, we need to determine the appropriate boundary conditions. The external forces applied to the mirror 's surface appear i n these boundary conditions in equations (3.1) and (3.2). In equil ibr ium, the external force, P , on a unit area of surface, df, of the mirror must be balanced by the force, aikdfk, of the internal stresses acting on that element (i.e., P{df — <Jikdfk = 0). Wr i t i ng dfk = nkdf, where n is a unit vector along the outward normal to the surface, we have o-iknk = P{. (3.3) This set of boundary conditions must be satisfied at every surface element of the mirror. Recal l ing that arz = azr, and since Pr and Pz are much smaller than the internal stresses of the deformed cell, we can expand (3.3) to 40 aTTnr + o~rznz = 0 (3.4a) azrnr + azznz = 0. (3.46) Let us now examine the boundary conditions at the three external mirror surfaces: upper, lower, and circumference. From Figure 12, we see that nr = — ns in# and nz — i icos#. Using these equations in (3.4) gives °~rz = o~rrt&n9 = azz cot 9. (3-5) For (3.5) to be val id for a l l 9 (specifically 9 = 0), we must place the following restriction on arz and azz at the upper surface of the mirror cell: o-rz(u) = azz(u) = 0. (3.6) A n identical argument for the lower surface, again using Figure 12, leads to the addit ional boundary conditions: °rz(t) = 0zz(t) = 0, (3.7) while the boundary conditions for the outer circumference are simply: arr(R) = arz(R) = 0. (3.8) We can use these boundary conditions to reduce the equi l ibr ium equations, (3.1) and (3.2), to a solvable form. Integrating (3.2) over z, from the lower (t) to the upper (u) surface of the mirror cell, gives 1 3 r r r~drr J  C r z dZ + a z z ^ ~  a z z ^ ~ J P9dz = ®, (3-9) 41 where ozz{u) = crzz(£) = 0, from (3.6) and (3.7), and f^pgdz = P, where P is the weight per unit area of the mirror cell and mercury, i.e., i 8 r P = - — r J arzdz. (3.10) M u l t i p l y i n g the first equil ibr ium equation by z and integrating z over the same range yields 1 d fu fu dc 1 fu - — r / arrzdz+ / _rz z dz / osszdz = 0. (3.11) rdr Je ^ fe dz ^ r3 Je (a) (b) (c) Integrating (b) by parts, we find —^-zdz = [zarz\i — I arzdz = - / arzdz. (3.12) o To evaluate integrals (a) and (c), i n (3.11), we take advantage of the fact that the modulus of elasticity, E, is at least three orders of magnitude greater i n the upper and lower skins than i n the foam itself. Therefore, we can neglect the small contribution of the foam interior to these two integrals; i.e., assume that arr is constant i n the skin and zero i n the foam. Wr i t i ng (a) in terms of the definite (Riemann) integral, we find PU j—h/2+t rh/2+t / arrzdz = arr(£) j z dz + 0 +cr r r (u) / zdz, (3.13) Je J-h/2 Jh/2 ' Foam ' where h = h(r) is the thickness of the mirror cell at some given radius r, and t is the thickness of the mirror "skin" (see Figure 10). i.e., i; ht arrzdz « — [arr(u) — arr(£)], (3-14) 42 However, when the mirror is bent, it is stretched at some points and compressed at others, so there exists a "neutral" surface midway between the upper and lower surfaces, above which stresses are "compressive", and below, "extensive". These deformations have opposite signs above and below the neutral surface (Landau and Lifshitz 19886, p.38), so o~rr(u) w —arr(£), and equation (3.14) can be rewritten r I arrzdz ~ htarr(u). (3.15) Following an identical mathematical argument, integral (c) of equation (3.11) can be evaluated to give J a^zdz hta ^ (u). (3.16) Now, substituting (3.12), (3.15), and (3.16) back into (3.11), mul t ip ly ing through by r , and using (3.10), we find d hi f — [rhtarr(uj\ -assiu) = r arzdz: = / Prdr, (3.17) or rl Je J Incorporating the definitions of arr and a^, from (^4.11), into (3.17), yields d rrhtE ,dur a dur a u r x i htE ,ur a dur a ur. d r L l + <7V dr I-2a dr 1 - 2a r n l + <r V r 1 - 2a dr l - 2 a r ' = J Prdr, (3.18) which can be simplified to ^-[rh^\ + - }ur = D / Prdr, (3.19a) dr dr 1 — a dr r J + 43 where For small deflections of the mirror, the vertical flexure, uz, is related to the radial extension, ur, by _ dur d V The derivation of (3.20) can be found in its entirety in Landau and Lifshitz (1986, p.39). The solution of equations (3.19) and (3.20) provides the desired flexure as a function of the mirror 's radius. A n analytical solution can be derived for h = h0 = const, and is given below. (Note: i n general, the two equations must be solved numerically when h = h(r).) For h = ha, equation (3.19) becomes: K ^ [ r ^ ] - — u r = DPQ jRrdr, (3.21) Or Or r Jr where the external pressure, P, has been assumed constant, i.e., • - ' - S r + ^ - ^ ^ ^ - A (3.22) which is an Euler differential equation wi th general solution of the form DP 1 r 3 ur = c i r + -f + ^ L R M l n r - - ) - j], (3.23) where c\ and c2 are constants. For ur to be finite at r = 0, we demand that c2 = 0. i.e.. DP 1 r 3 U r = C i r + u : [ R 2 r { l n r ~ 2 ) - T ]- (3-24) 44 B y substituting (3.24) into (A.11a), an explicit solution for a r r is possible. Imposing boundary condition (3.8) (i.e., arr(R) = 0), we find that the constant c\ i n (3.24) is given by — DP R2 ci = -j^-[R2\nR-(l + 2a)^}. (3.25) Substi tut ing (3.25) into (3.24) now gives the equation for the radial extension of the mirror cell under mercury loading, function of r: ur = ^l[R2r\n^-r-(R2 + r 2 ) + ^ t f r ] . (3.26) We can then find the desired relationship for the vertical flexure, function of mirror radius, r, for a mirror of uniform thickness, h0, by coupling (3.26) w i th (3.20). i.e., whose solution is U z = ~8hT1 R~lr + ^ 4 R r ]- ( 3 - 2 8 ) A t the mirror 's edge (i.e., r = R), the flexure is given by •DP U z ( R ) = " 6 4 ^ ( 4 < 7 _ 7 ) j R 4 ' ( 3- 2 9 ) which, using (3.19b), can also be wri t ten ( l + q)(l-2cr)(7-4g) P o P 4 - 6 i ( W ) ^ 2 - ( 3 - 3 ° ) For both styrofoam and Kevlar , Poisson's ratio is approximately a ~ 0.2, so 45 uz{R) » 0 . 0 8 7 ^ . (3.31) The U B C / L a v a l L M T has the following specifications: R = 1.35m; hQ ~ 0.17m; t ~ 0.5mm; E ~ 2.2 x 1 0 1 0 N / m 2 ; P 0 347 .47N/m 2 (assuming a 2mm layer of mercury, 0.5mm thickness of Kevla r on the upper and lower skins, and 0.17m-thick foam core). Substi tut ing these values in (3.31) predicts a flexure at the mirror 's edge of uz(R) « —0.32mm. The flexure of the mirror under its own weight (i.e., non-mercury loaded) can also be calculated wi th equation (3.31), but i n this case, P0 includes only the contribution due to the Kev la r skin and foam (i.e., P0 « —80.6382N/m 2 ) . A t r = R, the unloaded flexure is found to be ~-0.08mm. The main result of this flexure is a simple shift of the mirror 's focal length. For our / / 1 . 8 9 2.7m mirror, a flexure of -0.32mm at the edge theoretically leads to an increase i n the focal length of ~ 1 8 m m . Distortion There is a second order effect of the mercury loading on the cell: the departure of the cell from parabolicity, or "distortion". The general equation for the paraboloidal figure is given by (2.5) (i.e., z = r 2 / 4 / ) . Under mercury loading, flexure of the mirror is introduced, where uz i n equation (3.28) denotes the deviation of the surface from that figure. Therefore, we can write the equation describing the figure of the loaded mirror as z' = z + uz. (3.32) We define the departure of the mirror from parabolicity, A P , as A P = z' -Cr2, (3.33) where C represents the best-fit parabola to the actual figure, so (3.32), (3.33), and (3.28) lead to 46 A P = Tf - W1^ b 5 " 7 + ^ R V | - c , r 2 ' ( 3 ' 3 4 ) To find the max imum distortion, A P m a i , we must solve = 0, which gives Substi tut ing (3.35) into (3.34) gives the maximum distortion, APmax: Aft... = K^)2 - (3-36) Typical ly , this occurs at r/R ~ 0.5 (P. Hickson, private communication). For the U B C / L a v a l L M T , using this and the mirror parameters introduced earlier, we find that the max imum distortion is A P m a x ~ —0.04mm. This is, again, negligible i n light of the mercury layer's abil i ty to redistribute itself to maintain parabolicity (see Section Longitudinal Stability Another important aspect of the mirror cell is its oscillation frequency about a horizontal axis. Figure 13 shows the situation schematically; points C and P are the center of mass of the displaced mirror and the pivot point (i.e., centroid) about which the mirror oscillates, respectively. These two points are separated by a distance d. It is important to note that the center of mass in Figure 13 is taken to be at the edge of the mirror which has "dipped" below the equil ibrium position. This is because the mercury w i l l shift towards this point, although it is obvious that the center of mass wi l l never be at the extreme edge, but w i l l lie somewhere between this point and the middle of the top surface of the mirror (we assume the mass of the mirror cell to be negligible compared to the mass of the mercury). A g a i n , consider the oscillating mirror to be a pendulum, wi th a frequency of hori-zontal oscillation given by 47 "> = £ V 7 ( 3 ' 3 7 ) (from Hall iday and Resnick 1978); where i" is the moment of inertia of a circular plate of radius R, height h0, and mass M , about an axis perpendicular to the plate's symmetry axis (perpendicular to the page i n Figure 13), and passing through its centroid, P. I is given by I = ^M(3R2 + h20) (3.38) (Fenster and G o u l d 1985). Also , the torsional constant, K (Hal l iday and Resnick 1978), is defined as K = Mgd, (3.39) where d is the separation of C and P i n Figure 13. Substi tut ing (3.38) and (3.39) into (3.37) gives the result 1 / 3gd Figure 13 shows the simple relationship between d, R, h0, and 9, where 9 = a r c t a n i / r , t is the thickness of the bearing's air "cushion", and r is the radius of the air bearing, such that d2 = R2 + ( ^ ) 2 - 2 i A c o s ( 9 0 ° - 9). (3.41) For t ~0.01cm, and r ~20cm, one finds 9 ~ 0° .03 . Thus, wi th R ~132.5cm and h0 ~17cm, equation (3.41), gives a max imum value for d of ~132.8cm. Taking the other extreme (i.e., the center of mass does not shift to the mirror r im , but remains 48 near the center), we have d ~ /i 0 /2=8 .5cm. These l imits on d set corresponding l imits on ut using equation (3.40). Thus, the range in frequency is found to be 0 . 2^i / t ^ 0 . 9 H z . Th i s range lies above the frequency generated by the rotat ing mirror, ~0 .16Hz (from Table II), hence, the generation of such a frequency capable of upsetting the mirror is unlikely. 3.2.2 Liquid Mercury Surface In Chapter 2, we introduced some of the second-order effects on the paraboloid surface of rotating mercury; here we discuss their significance for the 2.7m L M T . The first such effect was that of the earth's curvature on the mercury's paraboloidal figure (Section 2.3.1). The necessary corrective terms to the fundamental parabolic equation of state, (2.4), are contained i n equation (2.12). The first corrective term (—r2/2r^) corresponds to an increase i n the primary's focal length of only ~ 7.8/mi. The second and higher-order corrective terms are negligibly small . Equat ion (2.18) is an analytical expression for the effect of the Ear th 's Coriolis force on the mirror 's figure. A s wi th the Earth 's curvature, the first corrective term of this equation implies only a small focus shift ( A / ~-0.481mm, a defocussing of ~ 10".5). The second and th i rd terms correspond to comatic aberration ranging from +2 waves (at 5500A) at the "north" r i m of the mirror, to -2 waves (at 5500A) at the "south" r im . The effect on the 2.7m mirror is negligible, and even for larger L M T s , can be corrected entirely by correcting lenses (Section 3.4.2). The fourth and fifth terms of (2.18) (a combination of astigmatism and field curvature) are only ~ 1.7 x 1 0 _ 8 m m (~ A/32000 at 5500A) at the north and south rims, and are totally unimportant, as are a l l higher order terms. Should the symmetry axis of the mirror be slightly misaligned, we can predict the effect on the mirror figure from equation (2.26). Spirit levels allow one to align the symmetry axis w i th the local gravitational field lines to an accuracy better than 0 ~ 2". Assuming an error of this size, the first corrective term of (2.26) is a negligible comatic 49 aberration of ~750A (^A/7 at 5500A) at r = 132.5cm. The second-order and higher corrective terms are ~ 7.2 x 1 0 _ 1 1 m m at the mirror r im . More important are vibration-induced waves on the l iquid mercury surface (Sec-t ion 2.3.6), bo th gravity waves wi th long wavelengths, and capillary waves wi th short wavelengths. For gravity waves, the characteristic damping time, r 3 , given by equation (2.35), was found to be proportional to u-i. To avoid inordinately large damping times for these waves, the resonant frequency of the mercury-loaded mirror must satisfy: < h f i - <2-34> A 2mm layer of mercury implies that the vt should be - ^ l l . l H z . Bor ra , Beauchemin, and Lalande (1985) suggest that capil lary waves w i l l be the dominant wave structure present on the mercury's surface. The damping time for pure capillary waves, r c , is reduced as i / - 4 / 3 (equation (2.39)), provided For waves of A = 2.5mm (within the range of "pure" capillary waves - see Section 2.3.6), this suggests that vi should be ^ 12.0Hz. For A = 1mm, say, vt should be > 18.9Hz. A n added benefit of having high ui is reduced amplitudes of any capillary waves present, since A oc t / - 1 / 3 (see equation (2.43)). P lac ing theoretical l imits on A is very difficult (Beauchemin 1985); subsequent optical tests w i l l be necessary to determine them. Even though capillary waves may propagate for many wavelengths they wi l l not disrupt diffraction-limited performance, so long as the amplitude is -£400A at A = 5500A (Schroeder 1987). 50 3.2.3 Mirror Support The mirror and air bearing are supported by a three-point mount composed of a wood and styrofoam core, covered by a fiberglass-epoxy resin laminate. The wood core is an essential component of the mount, due to its superior damping characteristics (Zorzi 1985). This is important since background noise can excite the resonant frequency of the mount, inducing low-amplitude concentric ripples on the mercury surface (Borra et al. 1984, 1985 and Section 1.2.3). The resonant frequency of the mount should be as far as possible from the peak of the background-induced vibrat ion spectrum; unfortunately, determining this value theoretically is non-tr ivial (see Section 3.2.2). Assuming that most noise sources w i l l be at low frequencies (e.g. passing automobiles), the high resonant frequency of the fiberglass-epoxy resin laminate (Ungar 1985) should at least avoid these. Combined wi th the inherent damping characteristics of the mount should reduce the effects of a l l noise sources to a level compatible w i th diffraction-limited optics (see B o r r a et al. 1989). A c t u a l on-site vibrat ion analysis w i l l be needed before we can better specify the background noise spectrum. The mirror cell itself rotates on an essentially frictionless, air-lubricated bearing driven by a pulley and mylar belt system, which is in turn coupled to a small synchronous motor controlled by a crystal oscillator-stabilized A C power supply (Note: Oil- lubricated bearings may be util ised in larger future systems because they are an order of magnitude cheaper.). A i r pressure is provided by a compressor-air dryer-filter-regular system. Six braces wi th bearings are situated just below the lower r i m of the cell, to provide support to the mirror i n the event of an accidental load displacement on the surface. 3.3 Instrument Support Structure The support structure (i.e., t r ipod) for the C C D at the telescope's prime focus must be very stable. Horizontal movement must be restricted to less than a pixel wid th 51 on the C C D (15//m for the Ford 2048 x 2048 chip). Gradua l vertical motions should be less than a few centimeters in range to make auto-focus compensation possible., The principle sources of horizontal displacements include the excitation of a t r ipod leg's resonant frequency, and the deflection of the C C D dewar under wind loading; the most important source of vertical shifts is the tr ipod's thermal expansion and contraction. 3.3.1 Resonant Frequency Let us determine the resonant frequency, ur, of a t r ipod leg. The effect of such resonances on the horizontal equi l ibr ium posit ion of the C C D is difficult to quantify absolutely, as w i l l be discussed at the conclusion of this section. Figure 14 shows a simplified schematic of one leg of the prime focus support structure. Consider a horizontal I-beam t r ipod leg, fixed at one end (point B ) , and supported at the other (point A ) . The deflection, y , of such a beam of length £, under a uniform load W, as a function of the distance x from point A (Ko lb and Bennett 1985), is y = i h E ~ U { U X Z " 2 X 4 ~ £ 3 X ) ' ( 3 - 4 2 ) where E is Young's modulus of elasticity of the beam's material; £ is the distance between points A and B on the t r ipod beam; and I is the beam's cross-sectional moment of inertia. The max imum deflection of the beam, y m a x , occurs at x w 0 .4215£ Thus, from equation (3.42), this yields W£3 ymax ~ - 0 . 0 0 5 4 — , (3.43) The load, W, is given by the mass of the beam mult ipl ied by the local gravitational acceleration, g, w i th an additional factor of sin<^ to allow for the incl inat ion of the t r ipod leg wi th respect to g; i.e., W = -pg sin <j>A£, (3.44) 52 where A is the beam's cross-sectional area; and p is the density of the t r ipod material. The U B C / L a v a l L M T support structure uses I-beam t r ipod legs of a luminum alloy. F r o m Spotts (1985, p.18), the moment of inertia, I, for such a beam cross-section is I = -^(bh3 - hh*), (3.45) where the dimensions b, h, & i , h\, and t are as shown i n Figure 15. The I-beam chosen for the 2.7m L M T has dimensions (h x b x t = 3" x 2\" x JJ:") (Atlas Alloys Catalog: p .A44); i.e., h = 7.62cm; b = 6.35cm; t = 0.47625cm; bi = 5.87375cm; hi = 6.6675cm. Thus, the moment of inertia of the I-beam is I ~ 89.0cm 4 ; its cross-sectional area is A ~ 9.22cm 2 . Using the alloy density of p ~ 3 .48g/cm 3 , Young's modulus, E ~ 7 x 1 0 1 1 dynes / cm 2 , the angle of the t r ipod leg from the gravi-tat ional normal, <f> ~ 15°, and the length between points A and B , £ ~280cm, equation (3.44) gives the load on the beam a s W ~ —2.28 x 10 6dynes. The max imum deflection of the beam under such a load, given by (3.43), is j / m a x ~ 43.4^m. The fundamental reso-nant frequency of the t r ipod leg can be estimated from the "pendulum" approximation. i.e., (3.46) For J/max — 43.4^m and (j) — 15°, equation (3.46) gives a resonant frequency of ~38 .5Hz. Translat ing this resonance into a resulting horizontal displacement of the C C D is non-t r iv ia l . The top end of the leg (i.e., point A i n Figure 14) is i n reality not "simply supported"; it is far more complex. The mounting plates, instrument adaptors, and the effect of the two adjacent legs, leads to a problem whose solution is beyond the scope of this thesis. A s we saw in Section 2.3.4 when attempting to quantify the amplitude of surface waves in the l iquid mercury, an absolute determination of the amplitude is not possible unt i l on-site tests are underway. A s far as the relative amplitude of the 53 transverse resonances i n the t r ipod, we can say that their amplitude decreases wi th higher resonant frequencies, so ensuring that the value of vr is as high as possible is of prime importance. 3.3.2 Wind Loading The second way to laterally displace the C C D is wind pushing against the detector housing. Let us consider a simplified version of the structure, where the detector is supported by two legs of length £, as in Figure 14. Now, let the detector housing, of surface area S, be subjected to a force F, due to a wind of lateral velocity, v, and density, p. F rom the definition of the drag coefficient, CD, this wind force may be wri t ten F = ±pCDv2S (3.47) (Landau and Lifshi tz 1959, p.179), where CD ~ 0.5, S ~ 600cm 2 , and p ~ 1.3 x 1 0 _ 3 g / c m 3 . For a lateral displacement of length a, there is an associated compression of length 6 i n one of the legs, as well as a corresponding extension i n the other two. For small displacements it is a straightforward geometrical exercise, from Figure 14, to show that for <f> = 15° a « 3.86, (3.48) where 6, from the definition of Young's modulus E, is given by T£ b = KA' <3-49> A is the cross-sectional area of the I-beam t r ipod leg; and T is the tension i n a beam due to its compression/extension. (Note: The tensions i n the two beams are assumed equal - i.e., T\—Ti = T). 54 Balancing the horizontal forces (i.e., the wind force, F, w i th the total tension, T i cos 75° + T 2 cos 75°) leads to the result F w ]-T. (3.50) B y combining the four previous equations, we find a single equation describing the lateral displacement of the detector, a, as a function of the wind velocity, v: 3.8pCDS£v2 a = EA • ( 3 - 5 1 ) For £ ~ 7 x 1 0 n d y n e s / c m 2 , S ~ 600cm 2 , £ ~ 560cm, p ~ 1.3 x 1 0 _ 3 g / c m 3 , CD ~ 0.5, A ~ 9.22cm 2 (cross-sectional area of the I-beam), and a typical sea-level wind velocity of v ~ 5m/s , equation (3.51) gives a ~ 0.3 / /m, again much less than the Ford C C D pixel size of 15/xm. Th i s value is an extreme upper l imi t , since a 6m high si lo/dome w i l l protect the L M T from strong external winds. To displace the C C D by one pixel , the detector housing would have to be directly subjected to a hurricane-strength wind of ~127km/hr . 3.3.3 Thermal Effects W h i l e resonance and wind loading are sources of horizontal shifts, thermal variations, A T , during the night w i l l produce an expansion ( A T > 0) or contraction ( A T < 0) of the alloy t r ipod legs and a vertical shift i n location of the C C D . The change i n leg length, A ^ , under a temperature change, A T , is given by d = ^ c e x p A T , (3.52) where £ is the leg length (£ ~ 560cm) and c e x p is the coefficient of linear expansion of the leg's a luminum alloy ( c e x p ~ 2.25 x 1 0 _ 5 / ° C ) . For a typical nightly variation of ~ —10°C, the leg w i l l change in length by A£ ~ — 1.3mm, which results i n a shift of the C C D ' s vertical posit ion along the optical axis 55 of ~-1 .2mm. Therefore, a temperature change of only —1°C results i n a defocussing of - 2".6 ( F W H M ) . A temperature-sensitive auto-focussing mechanism w i l l be incorporated into the L M T to compensate for such thermal effects. Section 3.4.4 discusses such a device in more detail. A n alternative way to handle these thermal effects would be to select a material for the t r ipod wi th a lower coefficient of expansion (e.g. wood core w i th a graphite coating, since c e x p for graphite is 40x less than for aluminum (Gela 1985)). There would st i l l be a need for auto-focussing, however, since a temperature drop of 10°C would cause a focal point shift of ~ —0.03mm, corresponding to a defocussing of ~ 0".6. A solution of this sort also represents a substantial increase in cost. 3.4 Instrumentation Four major components w i l l comprise the prime-focus instrumentation package: the C C D (and its accompanying dewar and electronics), corrector lenses, intermediate-band interference niters, and auto-focussing mechanism. 3.4.1 Detector The U B C / L a v a l 2.7m L M T w i l l use a Ford 2048 x 2048 C C D i n the T D I readout mode (Sections 2.2.3, 2.3.4, and 2.3.5 and references therein). This detector is a three-phase, MPP-ope ra t ed (i.e., inverted bias) device wi th extremely low dark current. Its fundamental characteristics have been discussed elsewhere (Janesick et al. 1989) and are summarised i n Table III. U n t i l the Ford C C D ' s controller and dewar have completed their in i t ia l tests i n late-1990, a G E C 385 x 576 C C D wi l l be used during the optical tests - see Section 3.5). A n interesting aspect of the detector design for the L M T is a thermoelectric ( T E ) cooling mechanism, as opposed to the standard cryogenic coolant (i.e., l iquid nitrogen 56 ( L N 2 ) ) . This is made possible by the low dark current of the Ford C C D s (Janesick et al. 1989). Since the Ford C C D s have an inherent read noise of R ~ 6 e _ / p i x e l (L . Robinson 1990, private communication), we can place an upper constraint on the total noise budget, JV, of, say, 7 e - / p i x e l . Since readout and dark noise contribute to total iV by N = \/D + R2 (Walker 1987), this imposes an upper l imi t on the dark current, D, of ~13 e~/pixel/nightly integration time per object, where the nightly integration time per object is ~128.93s (Section 4.2). Thus, D must be less than ~ 0.10 e - / p i x e l / s . This upper bound on D corresponds to a C C D operating temperature of — 52°C (using Figure 23 from Janesick et al. 1989), which is well wi th in the performance of many commercial T E coolers. A general discussion of refrigerating C C D s w i th T E coolers can be found i n Petrick (1987). Operat ing the C C D at this higher temperature allows the use of a much smaller dewar, reducing the obstruction and weight at the prime focus. Wi thou t the need to continually refill a dewar at the prime focus, uninterrupted operation of the L M T is much easier. The Ford C C D , read out by the T D I technique, w i l l produce close to 2 G B of data nightly at the continuous rate of ~ 6 5 k B / s (Section 4.2). To support the data acquisition, and control the telescope itself, an on-site Sun 386i/250 computer, w i l l be used. Images are to be stored on an Exabyte 8mm cartridge tape drive, capable of handling 2 .3GB of data per cartridge. Opt ica l disk technology is being considered for future systems. The software for data analysis has been developed in-house by P. Hickson (1989, private communication). It is UNIX-based and optimised to handle the vast quantity of T D I data expected from the L M T . 3.4.2 Corrector Lenses The pr imary mirror of the U B C / L a v a l L M T is a 2.65m / / 1 . 8 9 paraboloid. W h i l e free of spherical aberration, the image formed by a paraboloid does suffer from negative coma, positive astigmatism, and field curvature (Schroeder 1987). Therefore, a set 57 of corrector lenses has been designed by C . L . Morbey of the Domin ion Astrophysical Observatory which w i l l provide a "corrected" field of view of ~ 1 /2° , ideal for use wi th the large-format Ford C C D (21'07" * 21'07"). The design is similar to the Wynne wide-field, spherical lens, paraboloid field corrector (Wynne 1974), i n that it has three elements, although Morbey 's design includes two aspherical surfaces and a cemented triplet. Numerica l data for the corrector lens system is given i n Table I V . Pre l iminary correcting optics have been designed for anticipated lOm-class L M T s , allowing a corrected field of view of 15° v ia the use of computer-controlled, actively-stressed, moving mirrors (Richardson 1987; Richardson and Morbey 1988). 3.4.3 Filters A s the scientific goal of this project is to obtain spectral energy distributions (SEDs) of a large number of galactic and extragalactic objects (see Chapter 4), it was necessary to acquire a set of image-quality optical interference filters. S E D s w i l l be obtained over the course of the year by observing wi th a different filter on each photometric night. Following extensive numerical simulations (Hickson 1991, i n preparation), a set of 40 filters was decided upon, covering the spectral range 4000A to lOOOOA. Table V gives the central wavelengths and F W H M bandwidths of each filter. B o t h the central wavelength, A c , and the bandwidth, A A , increase logarithmically, \ogvc ranging from 14.87 to 14.48 i n steps of 0.01, whereas the bandwidths follow the rule A A / A C = 0.046. The bandwidth increases w i th A i n order to ensure that the filters are equally efficient i n the red as i n the blue. The filters have a physical diameter of four inches, equal optical thicknesses, and a flatness tolerance of A / 2 . 3.4.4 Focussing Mechanism There are four pr imary sources which can lead to vertical translations of the prime focus or the instrumentation package itself. In Section 3.2.2, we saw that the Ear th ' s curvature and its Coriolis force produce small increases in the focal length of the mirror, but these 58 are independent of time. However, the th i rd effect, thermal expansion/compression of the l iqu id metall ic mercury, changes wi th ambient temperature fluctuations. In Section 2.2.2, this was found to give shifts i n the mirror 's focal point of ~ —3.6//m, over a 10°C temperature range during a night, which corresponds to defocussing of ~ 0".08. The final source of focus changes is again the effect of thermal variations, but upon the a luminum alloy legs. A s we saw in Section 3.3.3, for a temperature drop of -10°C, the C C D / i n s t r u m e n t a t i o n package wi l l shift along the optical axis by ~1 .2mm. This implies very substantial defocussing (on the order of 2".6 for every 1°C change in the ambient temperature). To compensate for time-dependent focus variations, the L M T w i l l be equipped wi th a room-temperature thermistor bolometer to sense local temperature fluctuations. U p o n proper calibration of the changes in the mirror 's focal point as a function of A T , a measured change i n the thermistor's resistivity w i l l signal a two-way motor to drive the instrument support plate up or down the optical axis to the opt imal focus, at a rate of ~ 40/xm/s. The support plate w i l l have a range of motion of ~ 5 c m , w i th a posit ioning accuracy of ~ 10/im. A change of 10/um i n the focus position corresponds to a defocus of only about 0".2, approximately 1/10 of the angular diameter of the expected average seeing disk. 3.5 Optical Testing Opt ica l testing of the U B C / L a v a l L M T wi l l be done on-site. The necessary facilities for extensive optical shop testing are not available to us at U B C . The only optical test which w i l l be applied to the mirror before starting its sky survey is the Ronchi test (Cornejo-Rodriguez 1978) because of its relative simplicity and ease of interpretation. 59 Three variants of the Ronchi test may be performed: (i) long time integrations (i.e., ~6.35s - rotational period of the mirror) to provide a qualitative assessment of the full mirror; (ii) a series of medium integrations (i.e., ~0.8s - 1/8 of a rotation), which, when combined, w i l l yield the "true" average shape of the mirror and help to study "printthrough" (Borra , Content, and Boi ly 1988; Section 2.4); and, (iii) very short integrations or "snapshots" of the l iquid mirror 's figure to study self-induced surface turbulence (R. Content 1989, private communication). Tests w i l l be conducted on sufficiently bright stars (i.e., V"^$) which pass through the field of view of the telescope. A manual north-south translation stage for the Ronchi grating w i l l give some flexibili ty i n the selection of appropriate test stars. 3.6 Summary The central element of the U B C / L a v a l 2.7m / / 1 . 8 9 L M T is a Kevlar-covered foam cell which possesses min imal flexure under loading by a 2mm layer of mercury, of ~-0.32mm, at the mirror r im . Its high resonant frequency minimises the excitement of both gravity and capillary waves, as well as avoids the longitudinal instabili ty regime of < 1Hz. Sources which can alter the C C D ' s horizontal posit ion include the excitement of the tr ipod's resonant frequency and deflection under wind loading, giving a maximum deflection of ~ 0.3/^m, for a 5m/s wind, significantly less than the Ford C C D ' s physical pixel size of 15/im. Thermal variations can affect the C C D ' s vertical posit ion by ~1 .2mm ( A T = 10° C ) , an effect which wi l l be compensated by a temperature-sensitive auto-focussing mechanism. 60 Chapter 4 UBC/LAVAL 2.7m LMT: SCIENTIFIC PROGRAM 4.1 Introduction The eventual goal of this project is to demonstrate the potential of even larger L M T s for future cosmological surveys. Even this 2.7m prototype should be capable of one of the largest survey ever made of distant galaxies and quasars i n only two years of operation. Section 4.2 summarises the operational parameters of the L M T . Section 4.3 out-lines the survey planned wi th the instrument, specifying l imi t ing magnitudes, expected number counts of various objects, etc. Final ly , Section 4.4 describes the impact of such a survey on studies of large-scale structures i n the Universe, the spectral evolution of galaxies, possible quasar/galaxy correlations, and galactic structure. 4.2 Telescope Specifications Located near Vancouver, the observatory's latitude, £, is ~ 4 9 ° 0 3 ' 3 6 " . The estimated ful l-width at half-maximum ( F W H M ) seeing is 2".0, while the sky brightness is es-t imated to be 20 V m a g / a r c s e c 2 . The telescope has an / / 1 . 8 9 2.65m mirror rotating 61 wi th an angular velocity, u) ~0.99 rad/s, corresponding to a linear velocity at the r im of u(D/2) « 131.18 cm/s « 4.72 k m / h r . These and other related site and mirror specifications are listed i n Table II. The detector for the U B C / L a v a l L M T w i l l be a Ford 2048 x 2048 C C D , wi th 15 ^ m x l 5 / /m pixels. The quoted detector read noise, R, is 7 e _ / p i x e l ; the quantum efficiency, q, at 5500A is ~30%; the full-well capacity is of order 3 x 10 5 e _ (Janesick et al. 1989). For a 2.65 m telescope operating at / / 1 . 8 9 , the angle subtended by each pixel is ~ 0".62, giving a total field of 21'07" on a side. The corresponding image scale is 41.25 arcsec/mm. The east-west axis of the C C D determines the effective integration time for each object per night, t, (i.e., the time for each object to traverse the detector length, L), where t is given by t = L x sec£, (4.1) where £ is the observatory's latitude and L is measured i n arcsec. For the Ford detector, L — 21'07", and at the latitude of the telescope, the effective integration time per object per night is ~128.93s. This translates into a scan rate for the C C D , vsca,n = L/t ~ 9.83 arcsec/s. The instrument's data capture rate, DCR, is given by DCR = (^)Nrnb, (4.2) where 9p is the angle subtended by one pixel , Nr is the number of columns i n the north-south direction, and is the number of bytes generated per pixel . This leads to a remarkable data capture rate of 65.06 k B / s w 234.22 M B / h r , assuming n j=2 bytes per pixel . 62 The backup device for the Ford chip w i l l be a G E C 385 x 576 C C D . The spec-ifications of the two C C D s when mounted at the L M T focus are compared i n Table III. 4.3 The Survey 4.3.1 General Characteristics The main objective of the L M T scientific program is a spectrophotometric survey of al l objects, complete to a given l imi t ing magnitude, i n the large region of sky accessible to the L M T over the course of a year. Spectral Energy Distr ibutions (SEDs) w i l l be obtained through multi-filter imaging. The advantages of multi-filter spectrophotometry (i.e., the use of a set of filters (Section 3.4.3), one per photometric night, to construct S E D s over the course of the survey period) of faint objects include: (i) application to crowded fields and extended objects where dispersion techniques are difficult or impossible, (ii) the wide spectral coverage possible, (iii) sensitivity to emission-line objects, and (iv) the lack of apertures i n the observations. The details of this technique can be found elsewhere (P. Hickson 1991, in preparation). W i t h the Ford 2048 x 2048 C C D , in one year, a strip of sky 0° .35 wide centered at declination S = + 4 9 ° . 0 6 , the telescope latitude, is available to the L M T . This corre-sponds to a survey area, A = 0° .35 x 360° x cos49° .06 ~ 83.0 deg 2 . (The corresponding survey area of the G E C C C D is ~23 deg 2 ; see also Table III.) A n extensive literature search was undertaken to catalog as many of the known extragalactic objects i n this survey area as possible. A l lowing for precession effects, Table V I lists the galaxies wi th published spectroscopic redshifts included wi th in the L M T strip at epoch 1991.0. Table V I I provides a similar list , but precessed ten years ahead to epoch 2001.0. These lists w i l l provide a calibration for our photometric method 63 of redshift determination (discussed in Section Similar lists for galaxy clusters, quasars, and BI Lac objects are given i n Tables V I I I , I X , X , and X I . B y A p r i l 1992, the right ascension range 0 8 h 5 4 m to 1 2 h 4 2 m , should be surveyed by about half of the 40 intermediate-band interference filters, allowing publication of prel iminary results. Table X I I shows the accessible right ascension range on the first day of each month of 1992, governed by an estimate of the time of local astronomical twilight. The strip of L M T data w i l l first be subdivided into a series of frames, to be prepro-cessed and analyzed. Init ial analysis w i l l be the automatic detection of a l l objects wi th in each frame, where an "object" is defined loosely as a set of connected pixels above some local threshold (P i t te l la 1988). "In-house" object-detection software is currently being implemented. The algori thm combines aspects of several finding routines currently in vogue (e.g. COSMOS: Heydon-Dumbleton, Coll ins , and M a c G i l l i v r a y 1989; Thonisch, M c N a l l y , and R o b i n 1984. ROME: K o o et al. 1986; P i t t e l l a and Vigna to 1979. FOCAS: Jarvis and Tyson 1981. APM: Maddox et al. 1990b). Each of these references provides an excellent introduction to "finding" algorithms i n general; the reader is directed to any of them for more information on the subject. A basic introduction is also given by P i t t e l l a (1988). A l l the above algorithms have been intercompared on an identical calibrated field by El l i s (1990). 4.3.2 Limiting Magnitudes and Number Counts Galaxies To better quantify the performance of the 2.7 m L M T , we treat its theoretical abil i ty to detect distant galaxies through the use of a 250 A intermediate-band filter centered at 5500A. Following Walker (1987), the opt imum signal-to-noise ratio, S/N, w i l l be S/N = U q (4.3) y/nq + Nq + R2m 64 where m is the number of pixels over which the galaxy extends; n is the total number of photons from the source delivered to those pixels during the integration; Nsky is the number of background sky photons delivered to the same pixels during that time; q is the detector's quantum efficiency; and, R2 is the number of equivalent detected photons, corresponding to the system rms readout noise per pixel , R. Nsky is given by NSKY = J V 0 m . g A o l O ^ - ^ A A A r ^ ' t m (4.4) where NO^BAO ~1000 p h o t o n s / s / c m 2 /A (Walker 1987) for a zero-magnitude AO star at A=5500A; (j, is the background sky brightness (~20 V m a g / a r c s e c 2 for the L M T site); A is the effective collecting area of the telescope (51755cm 2 for the L M T , including obscuration by the prime focus support plate and tripod); A A is the filter band-pass (250A, i n this case); r is the instrumental efficiency ( r = mirror efficiency x filter transmission = 0.7 x 0.8 = 0.56); <f><j)' is the pixel area projected on the sky (0.3844 arcsec 2 , for the Ford 2048 x 2048 C C D ) ; t is the integration time (128.93s per object per night i n the L M T survey); and m , the number of pixels over which the galaxy is detected, is given by ( ^ G a l a x y ) / ( ^ ' ) - Assuming a standard galaxy angular diameter, 2 r G a i a x y = 2".5, leads to m=12.77 pixels on the Ford C C D . The final result is Nsky ~ 45857 sky photons per galaxy image. The number of source photons, n, delivered to the same m pixels i n the same exposure is n = A W A O 1 0 ~ ° - 4 V AAXrt (4.5) where the variables were defined for equation (4.4). Using the L M T values above yields n = 9.34 x 1 0 1 1 _ 0 - 4 V . This into equation (4.3), wi th q = 0.3 and R2 = 49 photons (for the Ford C C D ) , and Nsky = 45857, gives 2 80 x rnii-o-4V S/N = . Z m x i U (4.6) V2.80 x 1 0 n - ° - 4 V + 14383 65 Solving (4.6) numerically, the l imi t ing magnitude, V , for a S/N = 3 detection of a galaxy in a single exposure is V ~ 2 2 . 2 . For the assumed galaxy diameter of 2".5, this corresponds to a surface brightness l imi t of ~23.9 V m a g / a r c s e c 2 . (Note, however, that l imi t ing magnitudes can be increased by co-adding observations from different nights.) A l imi t ing magnitude of V ~ 2 2 . 2 corresponds to a B magnitude of ~23.1 . Tyson and Jarvis (1979) determined that the galaxy number-density, NG, at the celestial pole (b — 90° ) , brighter than a given integrated B (A=4400 A ) magnitude (in the range 17<B<24) , is given by l o g i V G « 0 . 4 1 B - 5 . 6 3 . (4.7) Therefore, for a B l imit of ~23.1 , we expect No ~ 6934 galaxies per square degree at the pole. To estimate the total number of galaxies accessible i n the survey area of 83.0 deg 2 , we must consider the specific strip of sky sampled by the L M T at declination S ~ 50° . F rom Figure 1 of B o r r a (1987), by sorting galaxies into bins 10° wide i n galactic latitude, applying a simple cosecant law for galactic extinction of the form A B = 0.5(csc6 — 1) mag, and weighting each b in by the amount of t ime the L M T spends observing each interval of galactic latitude, the total number of galaxies accessible (at a S/N of 3) to the L M T in one year is ~270000. Table X I I I gives the l imi t ing magnitudes and numbers of galaxies accessible to the L M T assuming different signal-to-noise ratios, filter bandwidths, and the Ford and G E C detectors. Quasars and Stars To estimate the l imi t ing magnitude of the L M T for quasars and other star-like images, we reapply equations (4.3), (4.4), and (4.5), but use the diameter of the seeing disk to calculate m , the number of pixels over which the point source is detected. For an assumed seeing of 2".0 F W H M , m = ( ^ | e e i d i s k ) / ( < ^ ' ) ^8 .17 pixels. Therefore the l imi t ing "stellar" (i.e., quasar) B magnitude, for a single exposure at S/N = 3, is ~23.4. 66 From K o o , K r o n , and Cudwor th (1986) (particularly their Figure 7), the number of quasars per square degree brighter than B=23.4, at the celestial pole, is ~250. B y allowing for galactic extinction throughout the L M T strip, as done for galaxies above, this corresponds to ~8600 quasars accessible i n the 83.0 deg 2 field. Table X I I I gives the expected quasar statistics for different S/N, filter bandwidths, etc. A t a l imi t of F m a s ~ 22.5, the L M T field w i l l encompass ~ 7.5 x 10 5 stars, based on the number counts of Jarvis and Tyson (1981), and Yoshi i , Ishida, and Stobie (1987). 4.3.3 Spectrophotometric Redshifts Al though gathering slit spectra suitable to measure redshifts for a statistically sig-nificant sample of faint (i.e., V ^ 2 0 ) galaxies is very time-consuming, such a sample is essential to discriminate between competing theories of the large-scale structure of space-time (Sandage 1988). The optical multi-filter technique proposed for our survey may represent a more efficient way to obtain these data. Opt ica l broadband colours can be measured much faster than spectra, and can i n principle be translated into redshifts of galaxies by a method introduced by B a u m (1962). The technique is based upon the "shifting" of the continuum of a galaxy, constructed from intensities at n bandpasses, wi th respect to a reference continuum, based on the average of the continua of nearby galaxies of the same class. The redshift is estimated from the amount of shift which minimizes the residual between the observed and fiducial continua. Ideally, fiducials must be prepared for every major morphological class of galaxy, plus a sample of stellar and Q S O spectral classes, since each has a very different spectral signature. Thus, the technique can also be used to assign a most-probable classification to each object. The accuracy of the redshifts determined i n this way is sensitive to the number, and bandwidths, of filters for the continuum fit. The more numerous and narrow the filters, the better the result (Koo 1985). Recent studies by L o h and Spil lar (1986), using only 6 intermediate-to-broadband filters have shown that redshifts accurate to wi th in 5-20% of the spectroscopic values are possible. Previous studies (e.g. B a u m 1962; Gunn 67 1971; Oke 1971; Davis 1973; Couch et al. 1983; E l l i s et al. 1985; and MacLaren , E l l i s , and Couch 1988) are affected by selection effects associated wi th their samples of only very bright cluster ellipticals, or only radio galaxies. Different colour-redshift algorithms have been suggested recently by Guiderdoni (1987), Butchins (1981, 1983), and K o o (1985). These techniques involve the use of three or four broad-band colours which are reduced to two-colour plots from which a redshift can be estimated depending upon the galaxy's location i n the two-colour-plane. B o t h the Butchins and K o o samples are unbiased i n their selection of galaxy types, and claim accuracies of ~10%. Guiderdoni (1987) reports an accuracy of ~ 2 5 % for a selection of fields centered upon clusters w i th a strong central radio galaxy. The Palomar redshift survey (Oke 1971; Wi lk inson and Oke 1978) wi th the M u l t i -Channel Photometr ic Spectrometer ( M C P S ) (Oke 1969) is a good example of the ap-proach planned for the U B C / L a v a l L M T . The M C P S was a 32-channel spectrome-ter (one photomultiplier tube per channel) covering 3100A-11000A. A t wavelengths A^5600A, bandpasses of 80Awere predominantly used, while at longer wavelengths, bandpasses of 160Awere the norm. The simultaneous monitoring of 32 photometric channels produced a low-resolution continuum spectrum for comparison w i th reference spectra from nearby galaxy samples. Redshifts accurate to 2-5% were obtained reg-ularly, an accuracy that should be matched, or exceeded, by our system, which has 40 filters, between ~ 4 0 0 0 A a n d ~10000A, w i th bandwidths ranging from ~ 2 0 0 A i n the blue to ~450Ain the red (Table V , and Section 3.4.3). Unl ike galaxies, there have been few published studies which measure quasar red-shifts from low-resolution spectrophotometry, although Drinkwater (1988) has shown that objective pr ism spectra can attain an accuracy of 1% at a redshift of ~ 2 (V<20), w i t h S/N as low as ~ 2 , and spectral resolution of 80A. W h i l e our survey wi l l have poorer resolution by a factor of 3-4, at a l imi t ing magnitude of V ~ 21 and a S/N of 10, we st i l l expect to derive redshifts for ~3200 quasars (Table X I I I ) , doubling the size 68 of the largest Q S O catalogs currently available. The expected accuracy is discussed by Hickson (1991, i n preparation). 4.4 The Impact of the L M T on Astronomy 4.4.1 Galaxy Spectral Evolution The characteristic redshift of galaxies at V ~ 2 1 (the nightly l imi t of the U B C / L a v a l survey) is z ~ 0.5, corresponding to a light-travel time of about 45% of the age of the Universe ( K o o 1985). Therefore, our unbiased sample of ~80000 galaxies w i l l be able to explore the effects of galaxy evolution over a significant period of t ime. The spectral evolution of galaxies is a very slow process. To z ~ 0.7, at least, the majority of galaxies appear to exhibit no, or l i t t le , systematic changes i n spectra wi th t ime (Bruzual 1986). In the best current samples (e.g. K o o 1985), the early stages of galaxy evolution/formation have not been seen. To properly study galactic evolution, we must first understand the stellar pop-ulations of nearby galaxies. Before one can perform K-corrections on the S E D of an observed galaxy, an understanding of the rest frame S E D s of nearby galaxies must exist. Th i s point has been stressed in the galaxy evolution studies of Bruzua l (19883, 1986). Observations i n the ultraviolet are part icularly important since at high redshift, a rest frame A 0 w i l l be seen by an observer at A 0 ( l + z) , so for a z = 1 galaxy, the U V flux at 2500A w i l l be shifted to the visible at 5000A. The average U V S E D s of nearby ellipticals is better specified than for late-type galaxies, like spirals and irregulars. Thei r U V S E D s show very large differences from galaxy to galaxy, and even from region to region wi th in a given galaxy. Smal l statistical fluctuations i n the number of stars included i n an image of a late-type galaxy can produce significant differences i n the shape of the U V S E D (Bruzua l 1986). This makes the generation of an average spectral template for late-type galaxies very difficult, at 69 least i n the ultraviolet. For galaxies up to z ~ 0.5, this is not considered a serious problem (Loh and Spil lar 1986; Yoshi i and Takahara 1989), but for more distant galaxies, the lack of a good U V template S E D can reduce the accuracy of spectrophotometric redshifts. A t redshifts of ~ 1 , the point at which significant spectral evolution is predicted by some theories (see Bruzua l 1986, and references therein), the S E D in the U V and blue should be enhanced by more active star formation. Therefore, techniques like that of L o h and Spil lar (1986) could misclassify an el l ipt ical at high redshift as a spiral at low redshift (Yoshi i and Takahara 1989). B y using 40 intermediate-band niters, as opposed to the 6 used by L o h and Spillar, this problem should be minimized, although quantifying its effect on our sample w i l l be one of the pr imary scientific goals of this survey. The generation of deeper galaxy luminosity functions (Bingel l i 1988; de Lapperant, Geller and Huchra 1989) wi l l be another bonus of the L M T survey. 4.4.2 Cosmology A review of a l l the aspects of observational cosmology that are relevant to a survey the size of the U B C / L a v a l project is beyond the scope of this work. The reader is referred to Sandage's (1988) review article for a comprehensive discussion of the observations required to discriminate between competing cosmological theories. Sandage describes the four classical tests of world models: (i) V(z) (related to AT(m)-test v i a Mat t ig ' s equations), (ii) m(z), (ii) 0(z), and (iv) the time-scale test. L o h and Spil lar (1986) applied the first test, the volume-redshift test, to their pho-tometric redshift sample, i n an attempt to determine the value of the deceleration pa-rameter q0- A s pointed out by Sandage (1988), the V(z) test requires a complete galaxy count in redshift space (including al l morphological types and surface brightnesses), or else precise corrections for sample incompleteness must be made to discriminate be-tween values of q0 between 0.02-0.5. Inflationary theory dictates that the Universe be flat ( § 0 = 0 . 5 , i f A=0) , while most observations suggest that q0 < 0.1 (i.e., open U n i -verse). 70 L o h and Spil lar (1986) found a value very close to that predicted by inflation, but that result has been questioned by Bahca l l and Tremaine (1988), because of L o h and Spil lar 's handling of spectral evolution and neglect of the effects of galaxy mergers on the galaxy luminosity function at high z. Nevertheless, it may be possible to correct these shortcomings i n future work. However, Yoshi and Takahara (1989) estimate that to discriminate between q0 = 0.02 and qa = 0.5 world models v i a the V(z) test at a redshift of ~0.4 requires a galaxy sample complete to a l imi t ing V magnitude of ~23.5, two magnitudes fainter than the U B C / L a v a l L M T survey. This survey was not intended to resolve the debate surround-ing the values of such fundamental cosmological parameters, but by showing that our instrument can yield useful data, we hope to provide a strong scientific justification for the development of larger L M T s . 4.4.3 Large Scale Structure The isotropy of the microwave background is evidence that the Universe was very ho-mogeneous at the redshift of the surface of last scattering (z ~ 1000), the epoch of recombination (Partridge 1988). To date, the largest extragalactic redshift surveys have demonstrated an overall rule-of-thumb: the size of the largest feature has always been l imi ted by the extent of the survey (for a review, see Huchra 1988). One recent example is the discovery of the "Great W a l l " i n the C f A 2 survey (Geller and Huchra 1989), having a size of 6 0 / * _ 1 M p c x l 7 0 / * _ 1 M p c (0 .5< / i< l ) ! A t the distance scales surveyed thus far (z ~ 0.05), the Universe is s t i l l very inhomogeneous. Bahcal l (1988) has shown that superclusters of galaxies show statistically significant clustering on scale lengths of ~ 1 0 0 f o - 1 M p c . However, using the A b e l l catalog of galaxy clusters, Postman et al. (1989) suggest that there is no definitive evidence for structure on the scale of 3 0 0 / i _ 1 M p c . A t even larger distances, the study of quasar clustering is s t i l l i n its infancy, but has produced some interesting early results. Clustering of QSOs has been observed, but only on small scale lengths ( ~ 1 0 / i - 1 M p c ) , similar to galaxies (Shaver, Iovino, and 71 Pierre 1989). There is st i l l no evidence of inhomogeneity of quasar populations at scales of 5 0 - 5 0 0 / i - 1 M p c (Osmer and Hewett 1989). A l l of the above results for QSOs are based on very small numbers, since only about 1000 QSOs are known. The U B C / L a v a l survey could, after two years, quadruple the total number of known redshifts for both galaxies and quasars. Our galaxy survey wi l l supply redshifts to a l imi t ing magnitude of ~21 (i.e., z ~ 0.5), a factor of ten deeper than the C f A 2 survey, allowing us to examine the clustering of galaxies and clusters on G p c scale lengths. The large unbiased Q S O sample w i l l address the large scale dis tr ibut ion of quasars , and provide the means to examine the correlations between quasars and foreground galaxies/clusters (Yee 1990; El l ingson 1990; Ell ingson et al. 1989). The study of the size, distribution, and content of voids (Rood 1988) i n the large scale structure is also important. Kirshner et al.(1987) have surveyed the "prototype" void i n Bootes, both photometrically and spectroscopically, to a l imi t ing V magnitude of ~17 . Serendipitously, the L M T strip passes very near to the center of this void, sampling a large area of the void to a l imi t ing magnitude of ~ 21, to complement the work of Kirshner et al. 's (1987). 4.4.4 Structure of the Milky Way In addit ion to extragalactic studies, there are other projects "closer to home" which can benefit from this L M T survey. One of the most obvious is a study of our own Galaxy ' s physical structure v i a deep star counts as a function of galactic longitude, £, and latitude, b. B o t h two-component (spheroid and th in disk) and three-component (spheroid, th in , and thick, disks) models of Galact ic structure are s t i l l debated, although the latter has gained favour in the last few years. For an overview of both models, see the review by Norris (1987). The simplest two-component models were first proposed by Eggen, Lynden-Bel l , and Sandage (1962), based on a kinematically selected sample of nearby, high-velocity 72 stars, and while their analysis suffers from severe selection effects, many of the conclu-sions i n this landmark paper st i l l hold true. Bahca l l and Soneira (1984) have renewed arguments for a "pure" two-component model of the Galaxy, claiming that it is the best fit to a l l the available multi-colour star-count data (but see Gilmore, Re id , and Hewett (1985) for compelling arguments to the contrary). The discovery of what may be a th i rd major component of the Galaxy, the so-called "thick disk" ("thick" i n that it is an order of magnitude wider than the ~200pc "thin" disk), by Gi lmore and R e i d (1983) has led to a revolution of sorts i n the modeling of the Galaxy 's structure. Subsequent papers by Gi lmore , Re id , and Hewett (1985), Gi lmore and Wyse (1987), and Yoshi i , Ishida, and Stobie (1987) have further refined the three-component model, claiming that it is necessary to properly fit the available data. Even the "pseudo"-two-component models of Searle and Z inn (1978) have been updated to pseudo-three-component models (Zinn 1985), w i th the introduction of a thick-disk globular cluster subsystem. U p to now, only relatively small datasets have been available to model . One of the largest surveys is that of Gilmore, Re id , and Hewett (1985), who observed two fields down to V ~ 1 9 , including a total of ~28000 stars. Yosh i i , Ishida, and Stobie (1987), also surveyed ~18000 stars down to V ~ 1 9 over 21.46 deg 2 near the north Galact ic pole. Gi lmore, Re id , and Hewett (1985) recognized that complete multi-colour surveys to a l imi t of V < 2 2 are the next important step to resolve the current questions. The U B C / L a v a l L M T survey wi l l provide just such a dataset: ~750000 stars down to V ~ 2 2 . 5 . A particular advantage of the L M T survey for M i l k y Way studies is that it w i l l sample a wide range of galactic latitudes ( — 15° + 70°; from Figure 1 of B o r r a 1987), whereas previous surveys were restricted to a few small isolated fields (see the list i n Gi lmore and Wyse 1987). The C T l group at the Universi ty of Ar i zona (see Section 2.2.3), has a similar Galact ic structure program planned for their Tucson-based transit telescope, which wi l l cover a wider range of latitudes: —28° ^ b ^ + 90° , however, their star counts w i l l only go as deep as V=19.5 , because of their smaller C C D . 73 4.5 Summary Once completed, the U B C / L a v a l L M T survey wi l l be the largest of its k ind , providing spectrophotometric redshifts, at a S/N = 10 (a value necessary for accurate redshift determinations), for as many as 80000 galaxies and 3000 quasars, a factor of three larger than present samples. W h i l e investigating effects of galaxy spectral evolution on automatic redshift and morphological classifications wi l l be the main scientific goal of the project, the large database w i l l be of interest to cosmologists and galactic dynamicists alike. 74 T A B L E I: Key Contributions in the Development of the L M T Name Date Location Description Reference E . Capocci 1850 - Italy (Naples) Concept Only e Buchan pre-1857 USA Concept Only a f H. Skey late-1850s England Unspecified Size 6 9 R.C. Carrington 1868 England (Frensham) Unspecified Size c h H. Skey 1872 New Zealand (Dunedin) ~0.35m (Lab) d i R.W. Wood 1908 USA (Baltimore) 0.18m (Lab) R.W. Wood 1908 USA (East Hampton) 0.51m f/1.7->f/3 (Lab) jh 0.51m f/9 (Field) B.A. McA. 1922 Chile (Chanaral) 15.24m f/5.6->f/12 k Proposal Only - Never Built E . F . Borra 1982 Canada (Quebec City) Concept Only I V.P. Vasil'ev 1983 USSR (Khar'kov) l m f/0.72 (Lab) m E.F . Borra 1983-+1986 Canada (Quebec City) l m f/1.6 (Lab) nopqr l m f/4.7 (Field) E .F . Borra 1983->1984 Canada (Quebec City) 1.65m f/0.89 (Lab) no E.F . Borra 1987 Canada (Quebec City) 1.2m f/4.58 (Field) gr E.F . Borra 1987— 1^989 Canada (Quebec City) 1.5m f/2 (Lab) st P. Hickson 1989—>present Canada (Vancouver) 2.65m f/1.89 (Field) UV E.F . Borra 1991 Canada (Quebec City) 2.5m f/1.2 (Lab) w Notes to Table I a N o published evidence found for Buchan's role i n the development of the L M T outside of a brief statement i n Sir D a v i d Brewster's diary (Gordon 1870). 6 No published evidence found for this instrument's existence outside that of Skey's c la im i n a letter to Nature (1874). c N o published evidence found for this instrument's existence outside that of Wood's paper (1909b). d Aper ture was estimated from Skey's claim (1872) that an observer's "head", at prime focus, blocks an appreciable amount of the mirror 's collecting area. e M a i l l y (1872) f Gordon (1870) 9 Nature (1874) h W o o d (1909b) 1 Skey (1872) ' W o o d (1909a) k Rigge (1922) ' B o r r a (1982) m Vasil 'ev (1985) 7 1 B o r r a et al. (1984) 0 B o r r a et al. (1985) p Bor ra , Beauchemin, and Lalande (1985) * B o r r a et al. (1988) r Content et al. (1989) 3 Bor ra , Content, and Bo i ly (1988) * B o r r a et al. (1989) u Gibson et al. (1989) v Gibson and Hickson (1990) w P. Hickson and E . F . B o r r a (1989, private communication) 76 T A B L E II: U B C / L a v a l 2.7m L M T : Site and M i r r o r Specifications Parameter Value Lat i tude Est imated Seeing ( F W H M ) Est imated Sky Brightness M i r r o r Diameter M i r r o r Obscuration Effective M i r r o r Area Focal Length Focal Ra t io M i r r o r Angular Velocity Telescope Efficiency (5500 A ) + 4 9 ° 0 3 ' 3 6 " 2".0 ~20 V m a g / a r c s e c 2 265 cm 6.16% 51755 c m 2 500 cm 1.89 0.99 rad/s->9.46 r p m ~70% 77 T A B L E III: U B C / L a v a l 2.7m L M T : Detector Specifications Parameter G E C C C D Ford C C D P i x e l Number 385 x 576 2048 x 2048 P i x e l Size 22/mi x 22/um 15/um x 15^m P i x e l Size 0".91 0".62 Detector W i d t h (NS) 5'49" 21'07" Detector Length ( E W ) 8'43" 21'07" Image Scale 41.25 arcsec/mm 41.25 arcsec/n Quantum Efficiency ~ 4 0 % ~ 3 0 % Read Noise 10 e _ / p i x e l 7 e _ / p i x e l F u l l We l l Capaci ty 5 x 10 5 e~ 3 x 10 5 e" Integration T i m e / O b j e c t / N i g h t 53.19 s 128.93 s T D I Scan Rate 9.83 arcsec/s 9.83 arcsec/s Da ta Capture Rate 8.34 k B / s 65.06 k B / s Tota l Survey A r e a 22.9 deg 2 83.0 deg 2 Prel iminary Survey Area 3.6 deg 2 13.1 deg 2 78 T A B L E I V : Pr ime Focus Corrector (31' Field) - 2.65m / / 1 . 8 9 Paraboloid Dimensions in mm A x i a l Clear Radius Separation Mater ia l Radius "10000.00 186.41 189.12 6325.03 116.21 c F l a t c F l a t d73.51 44.83 56.15 141.31 e F l a t e F l a t ' F l a t 4600.80 34.18 146.52 30.89 130.22 8.00 8.00 18.39 25.00 6.30 5.00 1.50 5.70 mercury air B K 7 air air air B K 7 air B K 7 F K 5 4 B K 7 air F S air 1325.00 118.21 107.18 68.67 57.63 40.92 40.17 37.47 31.27 26.23 25.62 24.73 24.40 22.49 Notes to Table I V a C C = -1.00000 6 C C = 1.16339 c Interference Fi l ter d C C = 0.3 6 Dewar Window f Ford 2048 x 2048 C C D 79 T A B L E V : Fi l ters - Centra l Wavelengths and Bandwidths ( F W H M ) (in A) A A 4044 186 4138 191 4235 195 4333 200 4435 204 4538 209 4643 214 4751 219 4862 224 4975 229 5091 235 5210 240 5331 246 5455 251 5582 257 5712 263 5846 269 5982 275 6121 282 6264 288 6409 295 6559 302 6712 309 6868 316 7028 324 7192 331 7359 339 7530 347 7706 355 7885 363 8069 372 8256 380 8449 389 8646 398 8848 407 9054 417 9264 427 9480 437 9701 447 9927 457 80 T A B L E V I : The 21'07" Strip Centered on (5(1991.0) = - Galaxies W i t h Published Redshifts + 4 9 ° 0 3 ' 3 6 " Galaxy Epoch 1991.0 Type Type V V , Name a 8 Ref. (cz km/s ) Ref. 5 Z w 20 00 26 25 49 04 37 — — 5720 9 U G C 2459 02 59 56 49 01 45 Sdm 1 2464 10 0725+4914 07 28 11 49 08 57 S B a 2 5887 9 N G C 2541 08 14 01 49 05 25 SA(s)cd 3 582 9 N G C 2684 08 54 16 49 11 37 S 4 2858 4 1129+4916 11 31 40 49 02 37 - - 9863 9 1129+4917 11 31 40 49 03 37 - - 9363 9 1130+4923a 11 32 05 49 09 31 - - 10097 9 1130+4923b 11 32 20 49 09 01 - - 11251 9 IC 708 11 33 26 49 06 24 E 4 9438 4 IC 709 11 33 45 49 05 30 - - 9549 9 IC 711 11 34 16 49 00 18 - - 9724 9 IC 712 11 34 19 49 07 36 S 5 10017 9 1133+4923 11 35 00 49 09 24 - - 11307 9 1133+4927 11 35 14 49 13 00 - - 10454 9 1133+4926 11 35 17 49 12 00 - - 9870 9 1134+4925 11 36 02 49 10 53 - - 11223 9 1134+4920 11 36 07 49 06 47 - - 9641 9 U G C 6713 11 43 55 48 53 15 - - 896 9 N G C 5448 14 02 34 49 13 12 Sa 4 1989 9 1 Zw 81 14 07 57 48 53 22 - , - 15480 9 1407+4916 14 09 02 49 04 24 - - 1973 4 M r k 490 15 46 07 49 01 31 - - 2680 9 K D G 508a 17 19 01 48 59 26 E 6 7253 9 K D G 508b 17 19 08 49 03 02 S B b 6 7185 9 1 Zw 200 17 51 39 49 01 28 E 7 21650 9 1816+4855 18 16 50 48 56 22 S 8 9980 9 81 Notes to Table V I 1. Ni l son (1973) 2. Karachentseva and Karachentsev (1979) 3. de Vaucouleurs and de Vaucouleurs (1967) 4. Huchra etal. (1983) 5. Peterson (1970) 6. Stocke, TifFt, and Kaf tan-Kass im (1978) 7. Zwicky and Zwicky (1971) 8. Zwicky etal. (1961-1968) 9. Palumbo, Tanzel la-Ni t t i , and Vettolani (1983) 10. Tul ly (1988) 82 T A B L E V I I : The 21'07" Strip Centered on (5(2001.0) = +49°03 '36 ' - Galaxies W i t h Published Redshifts Ga laxy Name Epoch 2001.0 a 8 Type Type Ref. v 0 (cz km/s ) v 0 Ref. 5 Zw 20 00 26 58 49 07 56 — — 5720 9 U G C 2459 03 00 38 49 04 06 Sdm 1 2464 10 0725+4914 07 28 56 49 07 41 S B a 2 5887 9 N G C 2541 08 14 45 49 03 34 SA(s)cd 3 582 9 N G C 2684 08 54 58 49 09 19 S 4 2858 4 N G C 2854 09 24 03 49 11 50 Sb 4 2732 4 N G C 2856 09 24 15 49 13 50 S 4 2638 4 1129+4916 11 32 13 48 59 18 - - 9863 9 1129+4917 11 32 13 49 00 18 - - 9363 9 1130+4923a 11 32 38 49 06 12 - - 10097 9 1130+4923b 11 32 53 49 05 42 - - 11251 9 M r k 178 11 33 31 49 13 48 Irr 4 239 9 IC 708 11 33 58 49 03 05 E 4 9438 4 IC 709 11 34 17 49 02 11 - - 9549 9 IC 711 11 34 49 48 56 59 - - 9724 9 IC 712 11 34 52 49 04 17 s 5 10017 9 1133+4923 11 35 33 49 06 04 - - 11307 9 1133+4927 11 35 47 49 09 40 - - 10454 9 1133+4926 11 35 50 49 08 40 - - 9870 9 1134+4925 11 36 34 49 07 34 - - 11223 9 1134+4920 11 36 39 49 03 28 - - 9641 9 N G C 5448 14 02 57 49 10 19 Sa 4 1989 9 1407+4916 14 09 25 49 01 34 - - 1973 4 M r k 490 15 46 25 48 59 41 - - 2680 9 K D G 508a 17 19 17 48 58 50 E 6 7253 9 K D G 508b 17 19 24 49 02 27 S B b 6 7185 9 1 Z w 200 -17 51 54 49 01 21 E 7 21650 9 1816+4855 18 17 06 48 56 37 S 8 9980 9 83 Notes to Table V I I 1. Ni lson (1973) 2. Karachentseva and Karachentsev (1979) 3. de Vaucouleurs and de Vaucouleurs (1967) 4. Huchra etal. (1983) 5. Peterson (1970) 6. Stocke, Tifft, and Kaf tan-Kass im (1978) 7. Zwicky and Zwicky (1971) 8. Zwicky etal. (1961-1968) 9. Palumbo, Tanzel la-Ni t t i , and Vettolani (1983) 10. Tul ly (1988) 84 A T A B L E VIII: The 21'07" Strip Centered on 6(1991.0) = +49°03'36" - Ga laxy Clusters 1 Cluster Name Epoch 1991.0 a 8 v © (cz km/s ) A b e l l 649 02 23 19 49 13 09 — 0655+4911 06 57 56 49 07 40 -0658+4908 07 01 25 49 04 28 -0704+4913 07 06 49 49 09 09 -0756+4915 07 58 31 49 08 18 -0802+4908 08 04 42 49 00 59 -A b e l l 626 08 10 00 49 13 43 -0812+4921 08 14 35 49 13 29 -0823+4903 08 25 39 48 54 57 -0847+4923 08 49 23 49 13 50 -0855+4922 08 57 58 49 12 28 - • A b e l l 755 09 11 55 48 59 53 -1000+4911 10 02 43 48 59 06 -1001+4912 10 04 01 49 00 04 -A b e l l 990 10 22 57 49 12 33 -1050+4923 10 52 19 49 09 55 -A b e l l 1167 11 07 39 48 55 41 -1116+4914 11 17 48 49 00 33 -1118+4916 11 20 29 49 02 31 -1131+4923 11 33 26 49 09 24 -A b e l l 1314 11 34 19 49 05 24 10223 1151+4920 11 52 44 49 06 19 -A b e l l 1409 11 53 38 49 07 19 -1156+4909 11 58 13 48 55 18 -85 T A B L E V I I I - continued Cluster Name Epoch 1991.0 a 8 v 0 (cz km/s ) 1208+4913 12 09 34 48 59 19 — 1212+4927 12 13 33 49 13 20 -1217+4929 12 19 07 49 15 21 -1219+4910 12 21 07 48 56 21 -1221+4915 12 23 00 49 01 22 -1223+4908 12 24 54 48 54 23 -A b e l l 1615 12 47 17 48 55 35 -1251+4910 12 52 58 48 56 39 -1325+4912 13 26 19 48 59 15 -1403+4912 14 04 40 49 00 15 -1416+4915 14 17 54 49 03 41 -1422+4911 14 23 29 48 59 53 -1424+4906 14 25 53 48 54 58 -1429+4915 14 30 34 49 04 08 -A b e l l 2169 16 13 51 49 08 51 -1634+4916 16 34 37 49 10 59 -1647+4908 16 48 30 49 03 46 -1713+4913 17 13 58 49 10 14 -A b e l l 2269 17 27 33 49 10 02 -1752+4914 17 53 09 49 13 34 -1917+4855 19 17 55 48 59 32 — Notes to Table V I I I 1 From:' A b e l l , Corwin , and Olowin (1989); Zwicky et al. (1961). 86 T A B L E I X : The 21'07" Strip Centered on 6(2001.0) = + 4 9 ° 0 3 ' - Ga laxy Clusters 1 Cluster Name Epoch 2001.0 a 8 v 0 (cz km/s ) 0655+4911 06 58 41 49 06 50 — 0658+4908 07 02 11 49 03 34 -0704+4913 07 07 34 49 08 11 -0756+4915 07 59 16 49 06 39 -0802+4908 08 05 26 48 59 15 -A b e l l 626 08 10 44 49 11 55 -0812+4921 08 15 19 49 11 38 -0830+4923 08 33 15 49 12 33 -0836+4925 08 39 44 49 14 10 -0847+4923 08 50 06 49 11 35 -0851+4925 08 54 41 49 13 20 -0855+4922 08 58 40 49 10 07 -A b e l l 755 09 12 36 48 57 24 -1000+4911 10 03 22 48 56 11 -1001+4912 10 04 39 48 57 08 -A b e l l 990 10 23 34 49 09 30 -1050+4923 10 52 54 49 06 43 -1116+4914 11 18 22 48 57 16 -1118+4916 11 21 03 48 59 14 -1131+4923 11 33 58 49 06 05 -A b e l l 1314 11 34 52 49 02 05 10223 1147+4929 11 49 17 49 11 59 -1151+4920 11 53 16 49 02 59 -A b e l l 1409 11 54 09 49 03 58 -87 T A B L E I X - continued Cluster Name Epoch 2001.0 a 8 v © (cz km/s ) 1208+4913 12 10 04 48 55 59 — 1212+4927 12 14 02 49 10 00 -A b e l l 1522 12 19 54 49 11 01 -1219+4910 12 21 36 48 53 02 -1221+4915 12 23 29 48 58 03 -1251+4910 12 53 25 48 53 24 -1325+4912 13 26 44 48 56 09 -A b e l l 1804 13 49 13 49 13 50 -1403+4912 14 05 02 48 57 24 -1416+4915 14 18 16 49 00 56 -1422+4911 14 23 51 48 57 10 -1429+4915 14 30 55 49 01 29 -A b e l l 2169 16 14 08 49 07 22 -1634+4916 16 34 53 49 09 46 -1647+4908 16 48 46 49 02 45 -1713+4913 17 14 14 49 09 34 -A b e l l 2269 17 27 49 49 09 34 -1752+4914 17 53 24 49 13 28 -1917+4855 19 18 11 49 00 39 -Notes to Table I X 1 From: A b e l l , Corwin , and Olowin (1989); Zwicky et al. (1961). 88 T A B L E X : The 21'07" Str ip Centered on 6(1991.0) = +49°03 '36 ' - Quasars 1 Object Epoch 1991.0 ymag z Name a 6 1219+491 12 21 52 48 55 03 18.86 2.325 4C 49.29 17 30 30 49 06 49 18.80 1.038 O Z 453.7 23 34 17 49 12 14 18.20 1.534 Notes to Table X 1 From: Hewit t and Burbidge (1987,1989); Burbidge and Hewit t (1987). 89 T A B L E X I : The 21'07" Strip Centered on 6(2001.0) = + 4 9 ° 0 3 ' 3 6 " - Quasars 1 Object Epoch 2001.0 y m a g z Name a 8 O J 448 2 08 32 28 49 13 10 18.82 4 C 49.29 17 30 46 49 06 23 18.80 1.038 Notes to Table X I 1 From: Hewit t and Burbidge (1987,1989); Burbidge and Hewit t (1987). 2 BI Lac object. K n o w n optical variable; average V m a g given. 90 T A B L E X I I : Right Ascension Observing Window (1992) 1 Date Start E n d P S T S T ( = a ) 2 P S T ST(=, J an 1 18:20 0.8 06:10 12.7 Feb 1 19:00 3.5 05:50 14.4 M a r 1 19:40 6.0 05:10 15.5 A p r 1 20:30 8.9 04:00 16.4 M a y 1 21:40 12.0 02:40 17.0 J u n 1 23:20 15.7 01:00 17.4 J u l 1 — — — — A u g 1 22:20 18.7 02:20 22.8 Sep 1 21:00 19.4 03:30 2.0 Oct 1 19:40 20.1 04:30 4.9 Nov 1 18:40 21.1 05:10 7.6 Dec 1 18:10 22.6 05:50 10.3 J a n 1 18:20 0.8 06:10 12.7 Notes to Table X I I 1 Pacific Standard T ime ( P S T ) and Siderial T ime (ST) are both approximate. Columns 3 and 5 give the in i t ia l and final right ascensions (governed by astronomical twilight) i n this L M T strip, accessible on each given night. 2 For zenith observations, the S T is equal to the right ascension, a. 91 T A B L E X I I I : Spectrophotometric Survey: Predicted Nightly Performance 1 Parameter G E C C C D Ford C C D I B 2 B B 3 I B 2 B B 3 S o N-o —=10 N N~6 —=10 N-o ^•=10 N —=10 N Saturating Stellar V m a s 12.3 12.3 13.7 13.7 12.7 12.7 14.1 14.1 L i m i t i n g Stellar V m a s 22.1 20.8 22.9 21.5 22.5 21.1 23.2 21.9 L i m i t i n g Ga laxy Integrated V m a g 21.9 20.6 22.6 21.3 22.2 20.9 22.9 21.4 Number of Galaxies Accessible 4 56000 16000 110000 32000 270000 79000 520000 130000 Number of Quasars Accessible 4 1800 600 3000 1300 8600 3200 14000 5800 Notes to Table X I I I 1 Assumptions: A l l results quoted for V band (5500 A ) ; typical galaxy diameter = 2".5; filter transmission = 80%. 2 IB = Intermediate Band filter (AA=250 A , at A=5500 A ) 3 B B = Broad B a n d filter (AA=900 A , at A=5500 A ) 4 Quoted object counts cover the entire survey area accessible by the L M T . Galaxy counts: Tyson and Jarvis (1979); Quasar counts: K o o , K r o n , and Cudworth (1986). Figure 1: Surface of a Rotat ing F l u i d . A l iqu id rotating about its symmetry axis, z, in a constant gravitational field, takes the shape of a paraboloid (thick solid line) upon reaching equil ibrium. In equil ibr ium, the components of the centrifugal and gravita-t ional forces parallel to the tangent at any point P ( r , z) , F c | | and F^y, respectively, must balance (see Section 2.2.1). 93 / / 0/ Figure 2: The Effect of the Ear th 's Curvature on the Surface of a Rota t ing F l u i d . Unl ike the geometry of Figure 1, the gravitat ional field line at P(r,z) is incl ined at an angle 0 to the symmetry axis, z, due to the curvature of the Ea r th . In equil ibrium, F c | | and Fg\\ must balance. The equi l ibr ium shape is no longer parabolic, although the deviation from parabolicity is negligible (see Section 2.3.1). 94 ! Figure 3: The Effect of A x i s Misalignment on the Surface of a Rotat ing F l u i d . M i s -alignment of the l iqu id mirror 's symmetry axis, r , w i th respect to the local gravitational field l ine, g , can generate a form of " t idal waves" on the surface of the mercury. The equi l ibr ium shape is no longer parabolic, al though the deviation from parabolicity is negligible (see Section 2.3.3). 95 Illumination Response (a) co = -1/4 -2 -1 (b) co = 0 -1 CO 0 n 0 n +1 +2 direction of motion +1 +2 -1 0 n +1 +2 •(c) co = +1/2 0 n +1 +2 Figure 4: East-West Response of C C D to a Delta-Function. The response to an input delta-function (thick solid line) at a pixel n along a column of the C C D is shown as a function of u, the offset of the delta-function from a pixel center when the C C D clocking occurs. Three examples are given: u = -1/4, 0, and +1/2 (see Section 2.3.5). 96 -4.0 -2.0 0.0 2.0 4.0 B -4.0 -2.0 0.0 2.0 4.0 e Figure 5: East-West Response of C C D to a Gaussian. The response to an input Gaus-sian, f(6) (5a), is described by the convolution of f(8) w i th the "triangle function", A(8) (5b). The acquired east-west image, 1(6), is shown i n (5c). The result is a symmetrical ~ 5 % elongation of the input Gaussian's F W H M , wi th a corresponding ~ 4 % decrease in its central peak-intensity (see Section 2.3.5). 97 C C D Figure 6: Star T ra i l Curvature. The curvature of star trails across the C C D at non-zero latitudes is shown i n projection. A measure of half the detector's field of v iew, b, and its latitude, £, yields an estimate for A , the max imum deviat ion of a t ra i l from the center of a C C D column (see Section 2.3.6). 98 Column 2 A Column 0 CO 0<co< 1 Figure 7: Definit ion of Star T ra i l Curvature Entrance Phase. The curvature of star trails across the C C D at non-zero latitudes is shown. The entrance phase, u>, is defined as the offset of the "min imum" of the t ra i l from a reference C C D column edge (see Section 2.3.6). 99 Figure 8: North-South Response of C C D to a Gaussian. The response to an input Gaussian, f(8) (5a), is described by the convolution of f(9) wi th the response function, h(9) (5b). The acquired north-south image, 1(8), is shown in (5c). The result is an asymmetrical ~ 9 % elongation of the input Gaussian's F W H M , wi th a corresponding ~ 9 % decrease i n its central peak-intensity. The central peak of the image has been shifted ~ 0".04 to the right (see Section 2.3.6). 100 S U R F A C E D E T A I L (1/1) Figure 9: M i r r o r C e l l Composi t ion. The core of the cell is high-density styrofoam, laminated wi th two layers of bi-directional Kevla r in an epoxy resin/glass microsphere matr ix . A final layer of spuncast, pure epoxy-resin covers the core. The reflective surface is a ~ 2 m m coating of l iquid mercury. The mirror itself is mounted on an aluminum base plate/central hub which is fixed to the air bearing (see Section 101 z ii • a Figure 10: M i r r o r Ce l l Flexure: Variables. Section provides an analysis of the flexure of the mirror cell as a function of radius r. Generally, the thickness of the cell, h, is a function of r . The analyt ical solution provided i n the text assumes that h is constant (i.e., h(r) = hQ). The Kev la r / epoxy skin has a thickness denoted by t. 102 X Figure 11: M i r r o r C e l l Flexure: Coordinates. Section provides an analysis of the flexure of the mirror cell as a function of radius r . T h e net external force acting at some point of the cell , f, is related as shown to the gravitational and centrifugal forces, g and a c , respectively. 103 Upper Surface Lower Surface r r Figure 12: M i r r o r C e l l Flexure: Boundary Condit ions. Section provides an analysis of the flexure of the mirror cell as a function of radius r . Uni t normals in the r- and z-directions for both the upper and lower surfaces of the cell are shown, as well as the unit normal perpendicular to each surface, ii. 104 • - - - — 1 * R » T p-ij ^ s ^ s s s ' r c p Figure 13: Longi tudinal Stabi l i ty of the M i r r o r Ce l l . A mirror cell of radius R and uniform thickness hQ sits on an air bearing which provides a "cushion" t. The mirror can oscillate about a horizontal axis through its centroid P , by accidental displacement of the mercury load. A n extreme case is shown i n which the center of mass of the mercury, C , has been displaced from the center to the edge of the mirror (see Section 105 CCD Concrete Pad Figure 14: T r i p o d Legs: Resonant Frequency. Each leg has been taken to be fixed at points B and C , and simply supported at point A , to aid i n the resonant frequency analysis (see Section 3.3.1). 106 i hj 12 i h j / 2 -« t>! /2 t bi /2 b Figure 15: T r i p o d Legs: Moment of Inertia. 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Yoshi i , Y . and Takahara, F . 1989, Astrophys.J., 346, 28. Zinn , R . 1985, Astrophys.J., 293, 424. Zorzi , E . S . 1985, in Mechanical Design and Systems Handbook (2nd Edi t ion) , ed. H . A . Rothbart (New York: M c G r a w - H i l l Book Co. ) , p.40.1. Zwicky, F . , Herzog, E . , W i l d , P., Karpowicz , M . , and K o w a l , C T . 1961-1968, Catalogue of Galaxies and of Clusters of Galaxies (Pasadena: Cal i fornia Institute of Technology). Zwicky, F . and Zwicky, M . A . 1971, Catalogue of Selected Compact Galaxies and of Post Eruptive Galaxies (Zurich). 117 Appendix A MIRROR CELL STRESS ANALYSIS Let us first introduce some of the tools necessary to solve the stress/strain analysis of the mirror cell composition described i n Section The coordinate system and some of the important variables are shown i n Figure 10. It is convenient to work wi th cyl indrical coordinates (i.e., x1 = r;x2 = (f>;x3 = z), i n order to take advantage of the mirror's symmetry properties. Unfortunately, the basic equations of elasticity (Landau and Lifshitz 1986) are wri t ten using Cartesian coordinates (i.e., x'1 = x; x'2 = y; x'3 = z); thus, it is necessary to introduce the familiar transformation equations for these two systems: x = r cos<j> r = y/x2 + y2 ( A . l a ) y y = rsinti> </> = arctan ( —) (A. lb) z = z (A.lc) Differentiating equations (A.l) yields the further transformations: 118 The metric tensor i n cyl indrical coordinates (Landau and Lifshi tz 1975) is 1 0 0' 9 i j = ( 0 r 2 0 ) = l /f lr 'J ' . ( A 3 ) 0 0 1 The second stage is to introduce the basic elasticity theory required to solve for the flexure of the mirror under mercury loading. For small deformations of the mirror, as shall be assumed throughout, the strain tensor, is given by uik « - Ufc-.*) 2 5 2 [ # " ^ " ^ ( A - 4 ) where the Christoffel symbols, T*k, defined i n terms of the metric tensor are rt 1 _tm,d9mi , dgmk , % f c , , . _ v 119 (note: The Christoffel symbols can also be defined in terms of the basis coordinates (see equation (A.18).) u is known as the "displacement vector", which describes the displacement of some point of the mirror due to the deformation under loading. The components of u are ur — ur(r,z); u<£ = 0; uz m uz(r) (^ -6) where ur, the radial extension of some point, is a function of both r and z. Due to the symmetry of the problem, there is no angular extension, u^. It is the determination of the axial compression (i.e., flexure) of the mirror, uz, as a function of r, which is the goal of this section. From (AA), the components of the strain tensor in cylindrical coordinates (Landau and Lifshitz 1986, p.3) can be written Uik = / d u r 1 / dUj, _ U± , 1 8ur \ 1 / dur , dur. \ \ I dr 2 V dr r "1" r d<j> > 2 V dz dr > \ \(du± _ «£ , 1 dur \ 1 duj, j j j , 1/1 du, , du4 s 2\ dr r "i" r d<f> > r d<f> r 2V r d<j> dz > V 1 /d u r , d u z \ 1/1 duz , duz \ 2 \ dz dr 1 2\r d<f> dz ) dz (A.7) Substituting (AA) into (A.7) leads to / & 0 + uik = 0 ^ 0 . (A.8) \ l ( § t + ^) 0 0 / The stress tensor, given in terms of the strain tensor (Landau and Lifshitz 1986, p.13), is Vik = z~^—(uik + , < J 0 ueegik) (A.9) 1 + a 1 — Zo~ 120 where a is known as Poisson's ratio. The component of the stress tensor is the z t h component of the force on a unit area perpendicular to the Xfe-axis. The components themselves are ( 0~rr 0~rtf> 0~rz Cfir CT^cf, o $z | , ( A 1 0 ) 0~zr 0~ z<$> Ozz which by substituting ( A 8 ) into ( A 9 ) , yields err = T — h r 1 + z r - C - ^ + — A l i a 1 + a or 1 - 2a Or r E rdur duz, orz = r ^ - ^ h r f + -gf ] (A.116) 2(1 + a)L dz dr Er2 ur o~ dur u 1 + a r 1 — 2a dr r E rdur du vzr = r ^ ^ [ ^ + ^ f ] ( A l l r f ) 2(1 -|- (7) dz dr Ea dur ur ° z z = n J L v i o~T + ~ ( A l l e ) (1 + (7)(1 — 2(7) or r <yT4> = 0 > r = c>z = ^ = 0 ( A l l / ) In equi l ibr ium, balancing al l internal stresses wi th external forces gives the funda-mental "equil ibr ium" equations (in vector notation): F + p ( g + a c ) = 0, ( A 1 2 ) f 121 where p is the density; g and a<. are the gravitational and centripetal accelerations, respectively. The centripetal acceleration of the mirror has a negligible effect on its deformation (Gagne and Ar r i en 1990), and w i l l be ignored subsequently. Following the notation of Figure 11, and using the definition of the force vector (Landau and Lifshi tz 1986, p.4), F = <r^, the equi l ibr ium equations (A.12), i n compo-nent notation, can be writ ten where g = -gk, ( A 1 4 ) and the contravariant derivative (Landau and Lifshitz 1975), afk, is defined as °ik = 9£m*ik;m. ( A 1 5 ) i.e., Now, recall the definition of the covariant derivative, o~ik-,f. where the Christoffel symbols, Tlk£, i n terms of the basis coordinates are i _ , T O dx* dx'n dx'P d 2 x ' m dx> Lk£ Lnpdx,m d x k Qxt 1" d x k Q x t d x , m i ^•l0) recalling that x' = Cartesian and x 1 = cylindrical . 122 In Cartesian coordinates, V™ = 0 (Landau and Lifshitz 1975, p.241), so: d2x'm dxi B y substi tuting the transformation equations, ( A l ) and ( A 2 ) , into ( A 1 9 ) , it is seen that a l l the Christoffel symbols for cyl indrical coordinates are zero, except Tj2 = T221 = pTl2 = r. ( A 2 0 ) This result (i.e., equation ( A 2 0 ) ) w i l l be used shortly. W i t h ( A 1 9 ) , we can now write the covariant derivative, equation ( A 1 7 ) , as do-jk d2x's dxm d2x's dxm aih'£ ~ dx* ~ dx^dx* dx" ° m k ~ dx*dx< dx" <7,'ro' ( } and thus, daik d2x's dxm d2x's dxm a i k ' k ~ dxk ~ dx'dxk dx's<Tmk ~ d*xk dx"*™' ( 2 ) Now, ( A 1 6 ) and (A.21) can be used i n conjunction wi th (A.13) to examine the three components (i.e., i = r,<f), and z) which w i l l yield the necessary equil ibr ium equations i n a form suitable to solve for the flexure, uz(r). Case I: i = 1 (i.e., i = r) Wri t ten explicit ly, equation ( A 13) states: alk = 911^ll;l+ 922<ri2;2 + 933ari3;3 = 0 ( A 2 3 ) where i i _ dan _ m darr ( A O A \ 9 <^ii;i = -Q-f - I i i<7mi - 1 n ^ i m = ( A 2 4 ) 123 22 1 1 fda12 9 a^;2 = ^ 1 2 ; 2 = ^ l " 1 1 2 ^ 2 ~ J- 22 c r lmJ (A.25) 33^ _ d°"l3 0 <7l3;3 — -Q-^ i-l3Crm3 ~ i-33^lm = Qz • (A.ZO) Using (A.24) , (A.25) , and (A.26) i n (A.23) yields the first equi l ibr ium equation: or dz r rA Id d(7rz r dr dz 0~<j><t> = 0. (A.28) Case II: i = 2 (i.e. <t>) a2k = 0 n ^ 2 1 ; l + / 2 ^ 2 2 ; 2 + i 7 3 V 2 3 ; 3 = 0, (A.29) where 0 U<T2i ; i = 0 2 2 c t 22;2 = 233cT23;3 = 0. (A.30) Pu t t i ng (A.30) into (A.29) shows that there is no equil ibr ium equation generated by the <j> component. Case III: i = 3 (i.e., i = z) aik + Ph = fi,11^3i;i + g22cr32]2 + g33cr33;3 - pg = 0, (A.31) where 124 0 n<73i ; i = da 31 dx1 — r ^ c r m l — r™<7 3 r d °~zr dr (A.32) 9220-32;2 = ^ 3 2 ; 2 = ~\ [^-J ~ T?2am2 - T ^ S m ] = ^ ( A 3 3 ) 33 „ _ daw dazz . . 9 cr33;3 - -^ - j - - 1 3 3 c r m 3 - I 3 3 cr 3 m = (A34) A g a i n , combining (A.32) , ( A 3 3 ) , and ( A 3 4 ) wi th ( A 3 1 ) yields the second equil ibrium equation: dazr dazz a r z -dr- + -dr + — - p 9 = 0- ( A 3 5 ) i.e.. l r i ^ ) + ^ r - p 9 = °- (A.3G) The next step is to solve the two equil ibrium equations ( i . e . (A28) and ( A 3 6 ) ) analytically to find uz(r\ the solution of which is shown in Section 125 


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