1. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 1 TMT M1 Segment Support Assembly (SSA) Preliminary Design Review (PDR) Volume-2: SYSTEM LEVEL CALCULATIONS (See last slide for Revision History) Pasadena, California October 24-25, 2007 Contributors to the development effort: from IMTEC RJ Ponchione, Eric Ponslet, Shahriar Setoodeh, Vince Stephens, Alan Tubb, Eric Williams from the TMT Project George Angeli, Curt Baffes, Doug MacMynowski, Terry Mast, Jerry Nelson, Ben Platt, Lennon Rodgers, Mark Sirota, Gary Sanders, Larry Stepp, Kei Szeto TMT Confidential The Information herein contains Cost Estimates and Business Strategies Proprietary to the TMT Project and may be used by the recipient only for the purpose of performing a confidential internal review of the TMT Construction Proposal. Disclosure outside of the TMT Project and its External Advisory Panel is subject to the prior written approval of the TMT Project Manager. * Note: HYTEC, Inc. merged with IMTEC Inc. in March 2007.

2. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 2 Outline Volume-2: System Level Calculations –M1 Segmentation –Segmentation Correction (for Variable Segment Geometry) –Budgets: Installation & Alignment Edge Gap Actuator Stroke Mass

3. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 3 SEGMENTATION (see Backup Slides for more detail) System-Level Calculations

4. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 4 Segmentation M1 Array Segmentation –Six identical sectors of 82 unique segments Unique hexagonal shape Unique optical figure Segments (PSAs) clock 60 deg between sectors x y x y x y x y x y x y x y x y x y x y x y x y B1 B2 C2 C1 AA A A x y PSA C-Sys. (XY) Actuator & AAP Locations (Dots)

5. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 5 Segmentation Scheme Segmentation Overview: –Hexagons on a curved surface cannot be equal and regular (with regular gaps) –Segment outlines are determined by projecting a hexagonal array onto the optical surface, resulting in irregular hexagons (varied size and shape): Constant gap segmentation –Segment size and shape variations are important for many reasons: Affecting: Optical performance, Hex-correction, Size of mirror blanks, Length variation of Mirror Cell Top Chord members, AAP adjustment range… –By stretching the Base Pattern in-plane before projection, we can affect these resulting characteristics. X M1 Z M1 Base pattern: regular hex array Scaled hex array Scaling rule project vertices and center onto optical surface, // Z M1 Vertices in optical surface Center (scaled) (R M1 )

6. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 6 Segmentation Scheme Segmentation studies : –Radial scaling of base pattern: –9 different objectives evaluated –“Rule 7” selected  = 0.165 Minimize blank size Other metrics also favorable when  = 0.165 Objectives evaluated: 1.Minimize Irregularity 2.Minimize Variation of Segment Area 3.Minimize Variation of Circumscribed Dia. 4.Minimize Cell top bar length range 5.Minimize SSA aligner range 6.Minimize Edge angle scatter (diffraction) 7.Minimize diameter of largest circumscribed circle 8.Minimize Max Pivot Shifts to rebalance Whiffletree 9.Minimize Max figure residual after correction Note: 1. Per Mast and Nelson Where:  is the scaling parameter R max is the largest vertex radii (before scaling) R is the radial coordinate of a point in the base pattern to be scaled R scaled is the scaled radius of the point in the scaled pattern. k is the paraxial radius of curvature of M1 All coordinates in the M1 system Scaling Rule 1 (Only radial scaling studied, other parameters might be used for scaling, such as Azimuth)

7. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 7 Nominal cell member length = 1.24m 1: Max. Irregularity (mm RMS) 5: Max In-Plane Alignment Range (mm) 6: STD Edge Angle Scatter (mrad) 2: Range of segment area (%) 3: Range Circum. Ø (%) 4: Range of Cell Bar Length (%) 7: Max. Circumsc. Diam. – 1.21m 8: Max WT Pivot Shift (mm) 9: Max Pivot Shift resid. (nm) Value of metric Value of tuning parameter  Segmentation Scheme Sensitivity of objectives:  = 0.165

8. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 8 Segmentation Scheme Implementation: –Scaling rule affects the following: Mirror shape and size –Coordinates of mirror vertices –Definition of optical origin and PSA coordinate systems (unique for each seg.) Mirror cell node locations –Mirror cell top-chord length & variation Position of AAP Post relative to Fixed Frame hole –Segmentation Database contains the following parameters: PSA origins and Coordinate Axes in M1 Coordinate system Mirror vertices –expressed in PSA and M1 coordinate systems Location of Segment Clocking Mark in PSA XY-plane (Arrow points to center of M1) Coordinates of AAP mounting pads on mirror cell top chord Best fit radius for AAP mounting hole in Fixed Frame –minimize adjustment range over 82 segments Location of edge sensor positioning fiducials in PSA coordinate system –Segmentation Database under Revision Control at IMTEC See backup slides for excerpt from Segmentation Database

9. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 9 SEGMENTATION CORRECTION (see Backup Slides for more detail) System-Level Calculations

10. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 10 Segmentation Correction Approach Segment irregularity and size variations would degrade optical performance if not compensated for Single support system design for all 82 types: –Adjusted for each segment geometry Correction approach: –Rebalance each whiffletree: Pivot-point shifts Analysis of each type required –Drill holes for whiffletree pivots in custom locations for each segment type Low cost, automated CNC operation Balance masses would raise part count and add mass –Analysis of worst case corrections Max pivot shift estimate: ~3.5 mm Axial RMS error increases ~ 10% (~1nm)

11. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 11 Segmentation Correction Analysis Calculate unit cases (1gz applied to distorted segment) –Size Variations (Grow and Shrink) –Clocking –De-center (X,Y) –Irregularity Seven postulated cases (an approximate set, not orthogonal) Unit effects isolated by subtracting 1g RMS (in quadrature) Results show that pivot shifts are effective at compensating for segmentation (Next Slide) –Residual RMS is acceptable –Magnitude of pivot shifts practical Note: This work was performed on the 1.2m segment –Results suggest the correction approach and have not been repeated on the 1.44m design.

12. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 12 Segmentation Correction Analysis Evaluated 12 Cases Shown (for 1.2m segment) –Conclusion: Pivot shifts very effective correction method 10.2nm for 1.44m segment

13. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 13 Segmentation Correction Analysis Hardware Implementation of Pivot Shifts Can shift Pivots Several mm in-plane

14. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 14 INSTALLATION & ALIGNMENT BUDGET System-Level Calculations

15. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 15 Installation & Alignment Alignment & Registration –Estimate segment position errors in-plane, clocking, piston and tip/tilt –Due to: Registration - Clearance and Repeatability PMA Assembly Errors - Tower to Optical Origin/Axes/Plane Fixed Frame Alignment Errors - At Targets Target to Fixed Frame Tower-Attachment Tolerances Surveying Errors - Measurement Uncertainty (TMT Project Responsibility) –Position error estimates are based on RSS of various effects –Requirements are: In-plane alignment:+/-0.200mm (0.400mm range) Clocking alignment:+/-0.200mm at vertex (0.400mm range) In-plane repeatability:+/-0.050mm (0.100mm range) Clocking repeatability:+/-0.050mm at vertex (0.100mm range

16. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 16 Installation & Alignment Alignment & Registration Inputs & Assumptions

17. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 17 Installation & Alignment Alignment & Registration Inputs & Assumptions

18. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 18 Installation & Alignment Alignment & Registration Results

19. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 19 Installation & Alignment Optical impact of these positioning errors –using Terry Mast’s sensitivities Conclusion: –Design is very close to meeting requirements Need relaxation of clocking requirements for alignment and repeatability –In-plane repeatability: 0.100  0.125mm –Clocking repeatability requirement: 0.100mm  0.225mm at vertex –Alignment clocking: 0.400  0.450mm at vertex

20. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 20 GAP BUDGET System-Level Calculations

21. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 21 Gap Budget - Excluding Seismic Nominal Segment-to-Segment Gap: 2.5 mm Random Gap Reducing Effects: –PSA Manufacturing and Installation Tolerances0.488 mmRSS Sum –Environmentally induced PSA motions 0.139 mmRSS Sum –Mirror cell deformations0.486 mmRSS Sum RSS = 0.702 mm –Actuation Segment tip/tilt de-center:0.754 mm Adjacent segments with full differential tilt Fault Condition – Controller or Human Error Linear Sum: 1.456 mm –Gap Margin = 2.500 mm – 1.456 mm = 1.044 mm Note: Linear sum of all effects gives 2.479 mm gap change –within budget See back-up slides for more details + Linear Sum

22. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 22 Gap Budget - Including Seismic Assume 3.0g seismic with segment motions out of phase by 22.5 deg [0.39 factor * 3.0 * 0.203mm (1g deflection)] = 0.238 mm Random Gap Reducing Effects: –PSA Manufacturing and Installation Tolerances0.488 mmRSS Sum –Environmental PSA motions (w/seismic)0.275 mmRSS Sum –Mirror cell deformations0.486 mmRSS Sum RSS = 0.744 mm –Actuation Segment tip/tilt de-center:0.754 mm Adjacent segments with full differential tilt Fault Condition – Controller or Human Error Linear Sum: 1.495 mm –Gap Margin = 2.500 mm – 1.495 mm = 1.005 mm Note: Linear sum of all effects gives 2.717 mm gap change –exceeds gap allowable (Segments may contact slightly during EQ.) See back-up slides for more details + Linear Sum

23. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 23 Gap Budget Gap Budget Summary: –Gap margin appears acceptable using RSS summation –Conservative linear summation shows little or no margin –No changes recommended

24. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 24 ACTUATOR STROKE BUDGET System-Level Calculations

25. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 25 Actuator Stroke Budget –PMA assembly errors are large terms, still being refined –Mirror Cell thermal distortion is significant TBD –5mm actuator stroke seems sufficient

26. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 26 MASS BUDGET System-Level Calculations

27. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 27 Mass Estimate Current design meets both fixed and moving mass limits: –Component sizing complete –Current CAD mass summary (no contingency included) ComponentMass (kg)Reqt. Whiffletrees16.0 Warping Harness4.2 Moving Frame12.6 Lateral Support Tower & Locks19.7 Fixed Frame36.2 Adjustable Attachment Points6.3 Fixed Mass61.4 Moving Mass35.5≤ 45.0 Total Mass97≤ 90 Mirror Segment153.0 kg

28. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 28 Acknowledgements Acknowledgements: The TMT Project gratefully acknowledges the support of the TMT partner institutions. They are the Association of Canadian Universities for Research in Astronomy (ACURA), the California Institute of Technology and the University of California. This work was supported as well by the Gordon and Betty Moore Foundation, the Canada Foundation for Innovation, the Ontario Ministry of Research and Innovation, the National Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, the British Columbia Knowledge Development Fund, the Association of Universities for Research in Astronomy (AURA) and the U.S. National Science Foundation.

29. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 29 BACKUP SLIDES System-Level Calculations

30. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 30 Primary Mirror Segmentation: Detailed Discussion Of Segmentation Analysis Credit: Eric Ponslet, IMTEC System-Level Calculations

31. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 31 M1 Segmentation Problem Define details of segmentation of M1 into hexagonal segments –M1 curvature and constant gaps leads to irregular hexagons Infinite number of ways to define irregular hexagons on M1 surface –Limit choices by applying a radial scaling rule to an initial, regular hexagonal base pattern, in projection Approach (3D) –start with regular hexagonal array in the XY M1 plane (“base pattern”) –use a scaling rule to distort array in plane Current rule has one adjustable parameter –extrude (//Z M1 ) distorted array into optical surface –consider shape of resulting segments as projected into local frames Implement gaps Calculate various metrics

32. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 32 M1 Segmentation Problem Tuning the scaling rule –Rule has one adjustable parameter –Parameter can be adjusted to achieve various goals –Tuning problem: What are some useful goals to pursue? What are the best adjustments of the parameter to achieve those goals? –Compromises…

33. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 33 Previous Work (1/3) Original work by others –TMT.OPT.TEC.06.025.DRF01: Excel spreadsheet with calculated coordinates of 1.2m segmentation patterns for three scaling rules (Larry Stepp)TMT.OPT.TEC.06.025.DRF01 –T. Mast and J. Nelson, “TMT Primary-Mirror Segment Shape,” TMT Report No. 58, TMT.OPT.TEC.04.001.REL01, November 2004. TMT.OPT.TEC.04.001.REL01 –L. Stepp, “Advantages and Disadvantages of Segment Geometries,” TMT.OPT.TEC.05.031.DRF01, December 6, 2005. TMT.OPT.TEC.05.031.DRF01 Initial presentations of HYTEC work –E. Ponslet, “Primary Mirror Segmentation: Issue with Rule #1,” TMT.OPT.PRE.05.087.REL01 (HPS-280001-0045), January 17, 2006 TMT.OPT.PRE.05.087.REL01 –E. Ponslet, “Primary Mirror Segmentation: Corrected Results,” TMT.OPT.PRE.06.004.REL02 (HPS-280001-0046A), January 30, 2006 TMT.OPT.PRE.06.004.REL02 Detailed report –E. Ponslet, “TMT Primary Mirror Segmentation Studies,” TMT.OPT.TEC.06.005.REL01 (HTN-280001-0007), June 7, 2006 TMT.OPT.TEC.06.005.REL01 Other relevant documents: –T. Mast, G. Angeli, and S. Roberts, “TMT Coordinate Systems,” TMT.SEN.TEC.05.016.DRF04, September 2005. TMT.SEN.TEC.05.016.DRF04

34. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 34 Previous Work (2/3) Based on earlier scaling work by Larry Stepp & Jerry Nelson –Two scaling formulations –Three candidate goals for minimization (tuning of  ) 1. Maximum irregularity of any segment 2. Range of segment area 3. Range of circumscribed diameter Introduced concept of Best Fit Regular Hexagon (BFRH) –Least-square fit performed in local XY plane (XY SEG ) –Minimizes RMS value of distances from vertices of segment to vertices of BFRH Adjust radius of BFRH (1 parameter) BFRH can be centered at O SEG or free to re-center (2 parameters) BFRH can be aligned with X SEG or free to rotate (1 parameter) –Residual of LSQ fit is a measure of irregularity Irregularity (and size variations) impacts performance (imperfect SuperHex correction)

35. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 35 Previous Work (3/3) Showed that –Both scaling formulations are equally effective / equivalent –All 3 goals can be achieved by proper tuning, using a single formulation Goal 1: maximum irregularity reduced by factor 11 (with BFRH rotation) Goal 2: range of segment area reduced by factor 86 Goal 3: range of circumscribed diameter reduced by factor 11 –Allowing rotation of BFRH results in large improvements Only a factor for Goal 1 –Re-center of BFRH has negligible impact Results in more complicated definition of XYZ PSA Abandoned to keep definition of center simple from value without scaling

36. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 36 Irregularity: Definition irregularity  RMS distance between vertices of actual segment and vertices of LSQ best fit regular hexagon with arbitrary center, radius, and clocking angle –irregularity = RMS of residual of fit –general definition includes 4 variables: decenter (X & Y), radius, and clocking angle X Y Best fit regular hexago n Segment outline in {XY} SSA plane d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 de-center radius clocking

37. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 37 Re-centering BFRH has Negligible Impact These results from Aplanatic Gregorian design with a=0.6m

38. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 38 Recent Changes and Additions (1/2) Modified definition of segment center –Was: mean of XY M1 coordinates of 6 vertices (after scaling, before gaps) –Changed to: scaled location of centers in base pattern –New definition is closer to BFRH center > re-centering now even less useful ~0.01mm difference in irregularity New, more exact calculation of circumscribed circle –Was: centered at origin of local frame –Changed to: center is optimized to minimize diameter –Difference is small Added representation of M1 cell –Cell nodes and Interface nodes Added representation of SSA-Cell interface –3 SSA support points per segment, at single location in local coordinate system Represent SSA side of interface X PSA Y PSA Centered circle Free circle

39. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 39 Recent Changes and Additions (2/2) Added six additional Tuning Goals 4.Minimize range of bar lengths in top layer of cell 5.Minimize range of SSA alignment system 6.Minimize width of diffraction spikes from segment edges 7.Minimize largest circumscribed diameter (blank/boule size) 8.Minimize magnitude of WT pivot shifts 9.Minimize residual figure error from WT pivot shift correction Repeated all calculations for Ritchey-Chrétien design and new segment size –Geometry and segmentation K=60m, k=-1.00095 a=0.716m, t=45mm, gap=2.4mm 6*82=492 segments instead of 6*123=738 –WH pivot shift data not available for larger segments Used sensitivities from a=0.6m – not directly applicable –Observations: Optimal tuning almost identical Conclusions unchanged Can base decision on old baseline (a=0.6m, AG)

40. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 40 Segmentation Patterns New baseline: 6×82, 1.432m segments

41. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 41 Coordinate Systems XYZ M1 / RθZ M1 –O M1 at apex of M1 optical surface –Z M1 along axis of symmetry of M1 optical surface, positive toward stars XYZ SEG –O SEG in M1 optical surface, at center of segment –Z SEG  M1 optical surface –X SEG in RZ M1 plane XYZ TEMP * (= XYZ SSA in TMT.SEN.TEC.05.016.DRF04) –O TEMP = O SEG –Z TEMP = Z SEG –X TEMP // XZ M1 plane XYZ PSA * –proposed as replacement for XYZ SSA and XYZ SEG in TMT.SEN.TEC.05.016.DRF04) –O PSA = O TEMP –Z PSA = Z TEMP –  (X PSA, X TEMP ) = rotation to BFRH –SSA uniquely located in XYZ PSA Coordinates of SSA features are invariant in XYZ PSA Suggest keeping only XYZ M1 and XYZ PSA as official systems –Possibly also XYS SEG if used by others * Not currently an official TMT coordinate system (TMT.SEN.TEC.05.016.DRF04, September 2005)

42. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 42 Defining “PSA” Reference Frame X M1 Z M1 Base pattern: regular hex array Scaled hex array X TEMP Z TEMP Scaling rule extend vertices and center into optical surface // Z M1 center XY M1 = scaling rule × center Outline in XY TEMP BFRH Vertices in optical surface Rotation about Z TEMP, from XYZ TEMP to XYZ PSA Z PSA Y PSA X PSA X TEMP Y TEMP BFRH

43. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 43 Defining/Positioning Hardware Defining Segment outlines 1)Begin with circular blank from polishing 2)Engrave fiducials into optical surface of polished segment Fiducials define location of XYZ PSA reference frame 3)Cut segment outlines relative to fiducials segment edges are straight lines in XY PSA plane (basic) segment side faces // to Z PSA (basic) 4)Final-figure segments relative to XYZ PSA Optical prescription described in XYZ PSA Defining M1-Cell to SSA interface coordinates All assembly tooling aligned to fiducials only Physical outline or vertices are never used as datum Coordinates of support points are identical in all segment types, when expressed in PSA frame Converting those coordinates back to XYZ M1 frame provides global coordinates of interface points

44. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 44 X M1 Z M1 X,Y PSA Z PSA Cell Nodes (3) Interface Nodes (3) SSA Supports (3) Segment Vertices (before gaps) H cell H SSA R SSA Cell Nodes and Cell-SSA Interface Cell Nodes –At given distance H cell behind 3 of 6 pre-gap vertices, along local normal to optical surface –Form irregular triangle whose geometry depends on scaling Interface nodes –At 1/3 along length of cell members SSA Supports –3 points, representing nominal centers of AAP adjusters –At H SSA, R SSA from O PSA (same for all segments) –H SSA and R SSA optimized to minimize maximum distance to interface nodes X PSA Y PSA

45. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 45 Cell Nodes and Cell-SSA Interface Cell Nodes Interface Nodes SSA Supports (“center” of adjusters) H SSA R SSA X PSA Y PSA Z PSA H cell Local Normal to optical surface

46. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 46 Cell Nodes and Cell-SSA Interface

47. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 47 MATLAB Segmentation Code Produces regular Hex base pattern in XY M1 –given segment size (a), and ID and OD of M1 (cropping) Scales the base pattern in radial direction –Scaling rule adjusts radial coordinate (only) of centers and vertices of base pattern –Adjustable parameter  (intensity of scaling) –Outermost vertex of array (at R max ) is unchanged by definition of scaling rule Extrudes segment centers and vertices into M1 optical surface Defines segment-local coordinate systems –Z // normal to optical surface at segment center Calculates coordinates of cell and interface nodes –Top layer nodes and interface nodes Implements gaps Calculates size and rotation angle of BFRH –Defines final local systems (XYZ PSA ) Establishes coordinates of SSA support points Calculates various metrics of resulting segmentation Produces outputs files (ASCII) –segment vertex coordinates in M1 or PSA system –Cell node coordinates in M1 system Creates various diagnostic plots –Distribution of metrics across array and corresponding statistical distributions –Segment outlines –3D plots of array and cell

48. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 48 New Segmentation Goals (1/2) 1.Minimize segment irregularity 2.Minimize variation of segment area 3.Minimize variation in segment size 4.Minimize range of lengths of top members of cell Avoid having to build different length members Metric = Max(L)/Min(L)-1, in percent 5.Minimize required range of SSA alignment system Minimize maximum in-plane (XY PSA ) distance between SSA support points and interface nodes Radial and depth location of SSA supports adjusted for best fit 6.Minimize width of diffraction spikes from segment edges Based on edge angles projected on sky (?) Minimize scatter of projected (into XY M1 ) angle of segment edges Sort edges into 3 groups of angles (~0º, ~60º, ~120º) Calculate standard deviation within each group (Std 0, Std 60, Std 120 ) Metric = RMS(Std’s) = √ 1/3 (Std 0 2 +Std 60 2 +Std 120 2 ) This goal is optimized without scaling (  =0): Std 0 = Std 60 = Std 120 = 0

49. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 49 New Segmentation Goals (2/2) 7.Minimize diameter of largest circumscribed circle Minimize size of glass boules/segment blanks Now using exact calculation of circumscribed circle (free center) 8.Minimize magnitude of WT pivot shifts (rough estimate) Whiffletree pivot shifts are used to fine-tune axial support to actual segment shapes Custom machining of pivot features for each segment type Pivot shifts require “real estate” in the hub region of whiffletree components Large shifts could be difficult to implement Estimates based on study of pivot shifts for various modes of segment shape variations Based on “Correction of Segmentation Effects by Shifting Whiffletree Pivot Locations” TMT.OPT.TEC.06.009.REL01 (Eric Williams, June 2006) TMT.OPT.TEC.06.009.REL01 Metric = maximum estimated shift at any WT joint, for any segment 9.Minimize residual figure error after pivot shift (rough estimate) Pivot shift is very effective, but not perfect Figure error after optimal pivot shift is slightly worse than nominal value Estimates based on study of pivot shifts for various modes of segment shape variations Based on “Correction of Segmentation Effects by Shifting Whiffletree Pivot Locations” TMT.OPT.TEC.06.009.REL01 (Eric Williams, June 2006) TMT.OPT.TEC.06.009.REL01 Metric = estimated value of √(RMS corrected 2 – RMS nom 2 ), where RMS nom and RMS corrected are the RMS values of the SSA-induced surface errors, for a nominal regular segment (for which the WT geometry was designed) and the actual segment, after correction via WT pivot shifts, respectively

50. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 50 Goals 8 and 9: Axial Support Pivot Shifts BFRH is optimally clocked  no residual rotation Correction for segment size (case 1) – using BFRH radius for size Requires pivot shifts up to 0.49mm per mm of radial growth Used nominal segment radius = mean(BFRH radii) (could have used midrange value instead) Leaves residual figure error up to 0.425nm per mm of radial growth Correction for Irregularity of segment (mean of cases 6 to 12) Requires pivot shifts up to 0.636mm per mmRMS of irregularity Leaves residual figure error up to 0.197nm per mmRMS of irregularity Mean = 0.197 nmRMS/mmRMS Mean = 0.636 mm/mmRMS Table from “Correction of Segmentation Effects by Shifting Whiffletree Pivot Locations” TMT.OPT.TEC.06.009.REL01 (Eric Williams, June 2006)TMT.OPT.TEC.06.009.REL01 Applicable to a=0.6m

51. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 51 Goal 5: Range of SSA Aligners Plots show relative locations of SSA supports (blue dot, center of blue circles) and SSA support nodes (red dots) –Radial spacing between range circles is 0.5mm No Scaling (  =0) Max distance = 5.6mm Optimized (  =0.167) Max distance = 1.9mm Support nodes shown MUCH closer to one-another than actual

52. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 52 Tuning Results for RC, a=0.716m Goal: Minimize: 1 Max(Irreg) 2 Rge(Area) 3 Rge(Ø) 4 Rge(L) 5 Rge(Adj) 6 Std(  ) 7 Max Ø 8 Pivot Shift 9 Fig. error Optimal tuning  = 0.243  = 0.122  = 0.160  = 0.161  = 0.167  = 0.000  = 0.165  = 0.230  = 0.141 Irreg. (mm RMS) % Relative to  =0 0.50 5 5.01 50 3.45 34 3.41 34 3.16 31 10.08 100 3.24 32 0.73 7 4.23 42 Area Range (%) % Relative to  =0 2.72 99 0.03 1 0.85 31 0.87 32 1.01 37 2.74 100 0.96 35 2.42 88 0.43 16 Circum. Ø rge (%) % Relative to  =0 1.35 52 0.65 25 0.22 9 0.22 9 0.26 10 2.61 100 0.24 9 1.16 44 0.35 13 Cell length rge (%) % Relative to  =0 1.44 53 1.35 49 0.95 35 0.95 35 0.97 35 2.75 100 0.96 35 1.36 50 1.13 41 SSA align. rge (mm) % Relative to  =0 3.02 48 3.27 51 2.29 36 2.27 35 2.18 34 6.46 100 2.19 34 2.85 44 2.78 43 Edge  scatter (mrad) n/a 4.12 - 2.07 - 2.71 - 2.73 - 2.83 - 0.00 - 2.80 - 3.90 - 2.39 - Max(circumØ) (mm) % rel. to  =0 ≠ rel. to  =0.165 (mm) 1450.6 98.85 +6.4 1449.7 98.79 +5.5 1444.3 98.42 +0.2 1444.3 98.42 +0.1 1444.2 98.42 +0.1 1467.5 100 +23.3 1444.2 98.41 0 1449.5 98.77 +5.3 1446.9 98.60 +2.8 Max pivot shift (mm)* % Relative to  =0 2.65 30 3.21 37 2.95 34 2.94 34 2.90 33 8.74 100 2.91 33 2.56 29 3.08 35 Max. fig. res. (nmRMS)* % Relative to  =0 2.08 73 0.99 35 0.94 33 0.95 34 0.99 35 2.83 100 0.98 34 1.85 65 0.90 32 * These metrics estimated using sensitivities calculated for a=0.600m 6×82, 1.432m segments

53. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 53 Tuning Results for RC, 492×1.432m Rotation, but no re-centering of BFRHNominal cell member length = 1.24m 1: Max. Irregularity (mm RMS) 5: Max In-Plane Alignment Range (mm) 6: STD Edge Angle Scatter (mrad) 2: Range of segment area (%) 3: Range Circum. Ø (%) 4: Range of Cell Bar Length (%) 7: Max. Circumsc. Diam. – 1.21m 8: Max WT Pivot Shift (mm) 9: Max Pivot Shift resid. (nm) Goals 8 and 9 estimated using sensitivities from AG/a=0.600 design Value of metric Value of tuning parameter 

54. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 54 Segmentation Summary Scaling effects are almost not affected by segment size Goals 4 and 5 are not achieved very effectively –AAP adjustment range and cell member length variations remain significant after tuning the scaling rule Metrics for both goals only reduced to 35% of their un-scaled values Residual AAP adjustment ~ 2.2mm (radial) Residual cell bar length variation ~ 12mm –Definitions of node and/or support coordinates could (should?) be generalized Current work based on extremely simple (and limited) definition of cell Requires better understanding of cell fabrication approach Goal 7 can help save glass –Saves up to 23mm on blank diameter Tuning for goals 8 and 9 only moderately effective –Pivot shifts and residual errors reduced to ~30% of their value without scaling –Estimated* pivot shift reduced from 8.7mm to 2.6mm Relatively flat between  =0.12 and 0.25 –Estimated* residual error reduced from 2.8nmRMS to 0.9nmRMS Negligible when added in quadrature with nominal error (~10nm) Still no obvious, compelling reason to pick one particular tuning? –Which goal is most important? Tuning range between  ≈0.16 and 0.243 provides various compromises –Most goals are optimized closer to left edge of that band Exception: irregularity * Sensitivity numbers obtained from study with a=0.6m

55. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 55 Segmentation Database Segmentation Database Excerpt (for information) TMT M1 SEGMENTATION DATA - Eric Ponslet - HYTEC Inc. - 06-Mar-2007 15:14:15 This is the master segmentation data file HDB-280001-0003_Draft_3 generated by FinalSegm.m Matlab program This file is under revision control, please find current revision level in file name Latest official version resides on TMT Docushare document server, under document TMT.OPT.TEC.07.006.REL01 Unless otherwise specified, all linear dimensions are in meters, and all angles are in degrees ----------------------------------------------- SECTION 1: CONTROL PARAMETERS AND STATISTICS ---------------------------------------------- 1A: M1 GEOMETRY AND SEGMENTATION DATA M1 Radius of curvature: k = 60.000 m M1 Conic Constant: K = -1.000953000 Base pattern hex diameter: 1.4320 m Inter-segment gap: 0.00250 m (1/2 gap applied all around every segment (including outer edges of array) Segment chamfer width (projected into XY_PSA): 0.00035 m 1B: SEGMENTATION PARAMETERS Scaling Parameter: alpha = 0.1650 (radial scaling = (1+alpha*(Rmax/k)^2)/(1+alpha*(R/k)^2) With rotation of PSA Without recentering of PSA 1C: NOMINAL SEGMENT SIZE Nominal segment diameter = 1.440000 m Best Fit Regular Hexagon (BFRH) statistics: Min BFRH diameter = 1.436839 m, or 3.16 mm smaller than nominal Mean BFRH diameter = 1.440312 m Max BFRH diameter = 1.443711 m, or 3.71 mm larger than nominal Location of AAP nodes in PSA system (at 90, 210, and 330 degrees about Z_PSA): Radius (R_PSA) = 0.418241 m Elevation (Z_PSA) = -0.368276 m 1D: CELL DATA Distance from optical surface to cell nodes = 0.37250 1E: SEGMENTATION STATISTICS M1 Inner Diameters Gapped segments (glass): min diameter = 1.44808 m Optical surface: min diameter = 1.44849 m M1 Outer Diameters Gapped segments (glass): max diameter = 15.00049 m Optical surface: max diameter = 15.00010 m

56. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 56 Segmentation Database Segmentation Database Excerpt (for information) Segment Irregularity (RMS): min = 0.112 mmRMS, at segment # 2 max = 3.245 mmRMS, at segment # 66 Segment area: min = 1.3409 m^2, at segment # 82 max = 1.3538 m^2, at segment # 2 spread (max/min-1) = 0.96 percent Circumscribed diameter: min = 1.44057 m, at segment # 55 max = 1.44406 m, at segment # 32 spread (max/min-1) = 0.24 percent BFRH Clocking angle: min = -13.8032 mrad, at segment # 82 max = -0.0000 mrad, at segment # 36 BFRH Radius: min = 0.71842 m, at segment # 66 max = 0.72186 m, at segment # 2 spread (max/min-1) = 0.48 percent Mean segment area = 1.35 m^2 Diameter of segment of mean area = 1.44032 m Mean diameter of BFRH = 1.44031 m Cell bar length: nominal = 1.24015 m (length of cell bar built on unscaled, planar array) min = 1.24870 m max = 1.26072 m range (max-min) = 12.017 mm spread (max/min-1) = 0.96 percent AAP Adjustments (if post centered on node): In plane min = 0.10 mm max = 2.19 mm Z_PSA min = -0.16 mm max = 0.16 mm Estimated WT pivot shifts: min = 0.98 mm, at segment # 2 max = 2.84 mm, at segment # 66 Estimated superhex correction residual: min = 0.36 nm, at segment # 28 max = 0.93 nm, at segment # 66

57. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 57 Segmentation Database Segmentation Database Excerpt (for information) ----------------------------------------------- SECTION 2: DEFINITION OF PSA COORDINATE SYSTEMS ------------------------------------------- DEFINITION OF PSA COORDINATE SYSTEMS - SECTOR A Origin of PSA Coordinate System given as coordinates of segment center (ctr) expressed in the M1 Coordinate System, in meters Segment center lies in the M1 optical surface Orientation of PSA frame in M1 frame given as coordinates of 1xPSA, 1yPSA, and 1zPSA unit vectors, expressed in the M1 system For sectors B through F, PSA Coordinate Systems are rotated about Z_M1 by 60 to 300 degrees 2A: SEGMENT CENTERS / ORIGINS OF PSA COORDINATE SYSTEMS (in meters) seg# X_M1(ctr) Y_M1(ctr) Z_M1(ctr) 1 0.000000000 2.505174820 0.052299152 2 1.084848967 1.879013530 0.039229897 3 0.000000000 3.756438643 0.117590151 ----End of Excerpt

58. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 58 GAP BUDGET DETAILS System-Level Calculations

59. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 59 Gap Budget – Excluding Seismic

60. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 60 Gap Budget – Including Seismic

61. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 61 Gap Budget Segment tip/tilt produces decenter: –Center of rotation currently at Z PSA = - 55.739 mm –Vertex location (due to curvature): Z PSA = +4.32 mm –Radial location of actuators: R = 531 mm –Full differential tip/tilt results in 0.377 mm decenter (0.754 mm gap loss, worst case): Assumes total actuator travel of 5mm Center of tip/tilt rotation Tip/tilt angle: 5 mm over 796.5 mm = 0.63% Lateral decenter at vertex 0.63% of (60.059mm) = 0.377 mm 15.1 mm axial edge motion 0.63% of 720 mm = 4.5 mm 45mm 796.5 mm (1.5*531) Radius to actuators = 531 mm 60.059 mm at vertex

62. TMT.OPT.PRE.07.057.REL01 HPS-280001-0105 – Volume-2 – October 24-25 2007 – Slide 62 Revision History 11/13/07 Post PDR Corrections –Slide-7: Corrected location of vertical line to correspond to 0.165 value.