NIRSPEC

UCLA Astrophysics Program	U.C. Berkeley	W.M. Keck Observatory

Samuel B. Larson	May 25, 1996

NIRSPEC Optics Design Note 2.02

Guider Camera

Introduction

Observations using NIRSPEC must be guided to ensure that the target does not wander on scales larger than the desired spectroscopic resolution. We cannot rely completely on the slit-viewing camera for this task because its field of view (46" x 46") is too small to guarantee capturing a suitable guide star. Therefore, a separate camera with a larger field of view must be provided.

Guider systems can be quite intricate and expensive. In this application, at least three motions may be required: one to scan along the available field in search of a star, one to rotate the camera to follow the star as the image rotates, and one to focus the image to compensate for the telescope field curvature. The extreme guiding accuracy imposed by the NIRSPEC science goals adds to the complexity of this instrument. Given our limited budget and sampower, we want to select the simplest design that meets our performance requirements.

This document describes the simplest design concept, justifies its performance capabilities, and presents the full optical design and layout of the system.

Performance Requirements

There are several requirements that our guider must satisfy. Finding a guide star must not be cumbersome, and once found, it must be held in the guider field for the duration of the observation. The odds of finding no stars suitable for guiding must be very low (around the 1% level). Finally, the centroid of the guide star must be calculated to a certain accuracy. The highest possible resolution of a spectometer is 1/20 of a critically or better sampled resolution element (the slit width), which is obtained by centroiding on the spectral features (J. R. Graham 1994, priv. comm.; D. F. Figer 1995, Ph.D. Thesis). If the atmospheric seeing is about the same size as the slit width or smaller, then changes in the position of the target (along the width) will introduce errors in the wavelength calibration. The requirements placed on the guider are thus dependent on the science being done and the size of the seeing with respect to the chosen slit width. For NIRSPEC, the nominal slit width is 0.4", so ideally the guider should be able to provide guide star images that can be centroided to 0.02".

Design Motivation

The drive for simplicity means eliminating as many moving parts as possible. Instead of imaging a fraction of the available field of view, the simplest guider images the whole field, eliminating the need for a scanning mechanism. Also, intelligent software can track the guide star as the image rotates rather than physically rotating the guide camera.

Any part of the telescope field of view not used by NIRSPEC can be used for guiding, but it is natural to center the guider field on the optical axis. This way, guide stars will never move out of the field due to image rotation; guide stars will be closer to the rotation axis, minimizing their movement; guide star tracking will be unaffected by optical distortions since the aberrations are rotationally symmetric; and the distance between the science object and the guide star is minimized. Because rotational symmetry is preserved, it is also possible to design guider optics that correct the telescope field curvature, eliminating the need for a focus mechanism.

The guider design is shown in Figure 1, which traces two inner and two outer annulus field points. A flat pick-off mirror is placed just outside the NIRSPEC window, with a hole in the center to stay clear of NIRSPEC's field of view. The flat mirror sends light to a concave spherical mirror, which collimates the beam and redirects it through the hole in the flat mirror. A commercial lens then re-
images the field onto a CCD. The available guide field becomes annular: the inner radius is no smaller than that which falls into NIRSPEC (23" x 2) and the outer radius is defined by the edge of the CCD. The corners of the CCD are not used since stars in these regions would fall out of the imaged area as the sky rotated. This reduces the size of the mirrors.

Field of View

We want the largest field of view possible to minimize the risk of not finding a guide star. Choosing a CCD plate scale that critically samples the smallest typical seeing disk of the guide star gives the maximum field of view allowable. From the R-band seeing histogram in NIRSPEC SLITS (J. R. Graham 1996), the smallest FWHM is 0.4" which demands a plate scale of 0.2"/pixel. The size of the field is now governed by the size of the CCD.

The f/ratio of the camera can now be calculated using the equation

This gives f/1.5 for 15 m pixels, and f/2.5 for 24 m pixels. This camera requires a fast beam and it is important to get a CCD with large pixels. This is at cross-purposes with the need for a large CCD because pixel size generally decreases with larger CCDs.

For a 1024 x 1024 CCD, the total field of view would be 3.4' x 3.4'. The outer radius of the guiding annulus is then 1.7'. NIRSPEC must receive a radius of 23" x 2 = 0.54', so if the inner radius of the guide annulus is set to 0.7' (to allow an oversized field into the spectrometer), the total available field of view for guide stars is p(1.7')2 - p(0.7')2 = 7.6 square arcminutes.

Now we can get a rough estimate of the required size of the optical elements to capture this field of view. The guider is very near to the telescope focal plane, whose plate scale is 1.38"/mm. Capturing a field 3.4' across would require mirrors about 200 mm in diameter, fairly large for high-quality mirrors.

Guide Star Probability

The guider field of view determines the limiting magnitude necessary for guiding to ensure a reasonable chance of finding a guide star. For a worst-case scenario we will assume the observations are toward the galactic pole, where star counts are at a minimum. Note that some directions such as the "dark cloud" near the galactic center may be even worse. The density of stars at the galactic pole is taken from Allen, Astrophysical Quantities, 3rd Edition, scaled down by a factor of 0.73 according to Soniera and Bahcall (1981, NASA CP 3374). The probability of finding zero stars brighter than a given magnitude is calculated by

where r is the density of stars and A is the field of view. Table 1 shows these probabilities for a 7.6 sq. arcmin area over a range of V magnitudes.

TABLE 1. Probability of Finding No Guide Stars.

Limiting V magnitude	r (stars/deg2)

at galactic pole	Prob. of finding no stars in 7.6 sq. arcmin
18	764	0.199
19	1157	0.087
20	1834	0.021
21	2308	0.008

Guiding down to 19th magnitude presents a 1 in 12 chance of not finding a suitable star at the galactic pole, which is a bit too risky. A 20th magnitude limit brings the probability down to 2%, which approaches an acceptable risk. We clearly must be able to guide accurately at least to 20th magnitude for this field of view to be acceptable.

Not included in these numbers are galaxies, which become prevalent at these magnitudes and can be used for guiding. Thus, the probabilities calculated above can be considered conservative.

Integration Times

Any guiding magnitude limit is reachable provided there is no upper limit on the integration times used. In this application integration times are limited because 1) the telescope open-loop tracking accuracy degrades over time, and 2) field rotation will cause the guide star to smear over the course of an exposure. Using the methodology found in Keck Observatory Report 164 (J. Nelson and J. Cohen 1987), this section calculates the integration time needed to centroid accurately on guide at the required V=20 limit defined previously.

The two-dimensional rms error in centroiding is given by the relation

where n is the total noise in the aperture used for centroiding, n is the number of electrons due to the star, and FWHM is the size of the seeing disk (King 1983, PASP 95, 163). The total noise is given by

where	= total system throughput (incl. atmosphere and detector q.e.) in e−/photon

t = integration time

Vstar = V magnitude of guide star

Vsky = V magnitude of sky per arcsec2

A = effective area used for centroid calculation in arcsec2

a = size of pixel in arcsec

D = CCD dark current in e−/sec/pixel

R = CCD readout noise in e−/pixel

N20 = 7100 photons/sec from a V=20 star hitting Keck aperture above atmosphere

Appendix 1 shows the centroid uncertainty (in arcsec) as a function of integration time for a V=20 star, assuming a total system throughput of 0.3, 0.7" seeing, the optimal aperture for centroiding 5.55 FWHM2 (King 1983), full moon sky brightness (V=18), dark current of 1 e-/sec/pix and readout noise of 25 e-/pix.

Using these values, centroid accuracies of 1/20 slit width (0.02") are reached with 4.7 second exposures. However, centroid errors are highly dependent upon seeing. Using 1" seeing, it takes 16.3 seconds to reach the same accuracy. Note that when the moon is not full, these exposures are dominated by read noise. It is therefore important to use a CCD package with low read noise.

These times should be considered approximate at best, but they demonstrate that accurate centroiding down to 20th magnitude does not require unreasonable integration times. There is one other characteristic of this design that affects centroid accuracy, which is explored in the next section.

Guide Star Smear

Because this guider design does not compensate for image rotation, guide stars will move across the CCD during the course of an exposure. This will degrade the centroid performance by 1) elongating the star, and 2) distributing the starlight onto more pixels (noise). The second effect is probably the dominant one. Models of this effect show that increasing the exposure time improves the centroid accuracy, even though it allows the guide star to smear across even more pixels. In effect, the increased signal wins out over the increased noise. However, as the exposure time increases, the centroid accuracy asymptotically approaches some value. For a given observing situation it may not be possible to guide to the desired accuracy.

The allowable exposure time is also limited, as the telescope tracking rate needs to be corrected at finite intervals. Exposures cannot be longer than the telescope can track on its own to the desired accuracy. Predicted Keck open-loop tracking accuracies are as follows: 0.02 arcseconds rms over 10 seconds, 0.1 arcsec over 10 minutes and 0.5 arcsec over 1 hour (J. Nelson and J. Cohen 1987, Keck Observatory Report 164). Actual tracking slippage has been measured at 4.5 arcsec over 1100 seconds (Beth Klein 1995, private communication), but the conditions under which this rate was measured are unknown. If this slippage is typical, it indicates that the telescope cannot go unguided for any length of time, which is impossible to satisfy since it takes several seconds just to read out the CCD, perform the centroid calculation, and send a tracking correction to the telescope. For the sake of argument, we shall set 10 seconds as the maximum allowable exposure time.

Guide star smear models were derived by calculating the noise per pixel from calculations described earlier, then estimating how many pixels the star will move across during the exposure time, given a star speed across the CCD. A total noise was then calculated by multiplying the noise per pixel by the square root of the number of pixels the star inhabits. This total noise was fed into the centroid error calculation presented earlier to get a final centroid error.

Star speeds were calculated using equations from V. P. Yegorov and Yu. A. Sabinin (1971, New Techniques in Astronomy, ed. H. C. Ingrao, Chapter I-8). The field rotation rate in rad/sec is

where A is the azimuth, Z is the zenith angle, is the local latitude (19.833° for Keck), and w0 is the sidereal rate (7.2925 x 10-5 rad/sec). This equation can be expressed in terms of hour angle and declination by using the following conversions.

where is the declination and t is the hour angle. The rotation rate (rad/sec) is multiplied by the star's distance from the rotation axis at the telescope focal plane (mm) and by the telescope's plate scale (arcsec/mm) to obtain the star's speed in field arcseconds per second.

Appendix 2 contains a velocity map around the zenith, the region of fastest image rotation, for a star on the outer edge of the guiding annulus (the worst position). Declination runs vertically and is defined by the apparent declination minus Keck's latitude of 19.833 . Hour angle (in minutes of time) runs horizontally.

Earlier it was found that for 0.7" seeing and full moon background, a V=20 star needed 4.7 second exposures to reach a centroid accuracy of 0.02". This is only true if the star is stationary. A 10 second exposure will still allow 0.02" guiding accuracy for star speeds less than 0.2"/sec. If the star resides at the outer edge of the guiding annulus, then from Appendix 2 we see that this speed is exceeded for declinations within ± 2 of zenith, extending at most 4.2 minutes on either side of the meridian. This means that near the zenith in this worst-case scenario, guiding could not be performed to this accuracy for as long as 8.4 minutes. Table 2 summarizes the size of this "poor guiding" region for seeing conditions of 0.4", 0.7" and 1.0" for a V=20 star in two cases: worst case (star on outer edge of annulus, full moon) and typical case (star in middle of annulus, moon half-illuminated), assuming 10 second maximum exposures.

TABLE 2. Region of Sky with Guiding Error Worse than 0.02" for V=20 Star
Seeing

(FWHM)	Worst Case	Typical Case
dec: 19.833° ±	hour angle	dec: 19.833° ±	hour angle
0.4 arcsec	0.3°	± 0.3 min	(in blind spot)	(in blind spot)
0.7 arcsec	2.0°	± 4.2 min	1.5°	± 3.0 min
1.0 arcsec	(all sky)	(all sky)	(all sky)	(all sky)

At 0.4" seeing in the typical case, the poor guiding region is completely encompassed by the Keck blind spot (± 0.22 dec within zenith, ± 1.4 minutes of hour angle). At 1" seeing, a 10 second exposure will not allow for 0.02" centroiding even with a stationary star. This is acceptable since 0.02" centroiding is only required during good seeing: poor seeing will either present a flat point spread function inside the nominal slit (making spectroscopy insensitive to target wander) or require a larger slit and thus a larger centroid error. For typical seeing (0.7"), even the worst case scenario makes guiding poor for only a few minutes for a very small range of declination.

Increasing the field of view to allow more V=19 stars only accepts stars further away from the rotation axis, leading to more smear. We cannot, for any reasonable magnitude star, completely eliminate the effect of guide star smear up to the Keck blind spot without physically rotating the guider camera. The issue, then, is whether the guider design be allowed to be driven by an area of sky this small. Eight minutes (maximum) is not much observing time to lose. It seems like a sufficient number of "ifs" have been built up in arriving at this worst-case scenario that an acceptable risk has been established.

Redder Bands

Until now, this note has assumed V-band guiding. Although it has been shown that satisfactory performance is achieved at V (0.55 m), redder bands such as R (0.65 m) or I (0.83 m) should be considered. The photon flux corresponding to a given magnitude decreases with increasing wavelength (20th magnitude fluxes are 7100 photons/sec @ V, 6000 photons/sec @ R and 5500 photons/sec @ I), but the star counts at a given magnitude increase considerably. This reduces the limiting magnitude necessary to ensure finding a guide star. The limiting magnitudes necessary to keep a 98% probability of finding a guide star at the galactic pole are V = 20, R = 19.5 and I = 18.6.

CCD quantum efficiencies between V and R are roughly the same, but at I the efficiency drops almost in half. Full moon sky backgrounds are much fainter at redder bands, and the seeing also improves. Even assuming no improvement in background, seeing or atmospheric transparency, the models used previously show that the integration times necessary to reach the desired centroid accuracy with a limiting magnitude star drop considerably with increasing wavelength. R-band exposures need only 65% of the V-band integration time, and I-band exposures need roughly 50% of the V-band time.

The problem with I-band is that expensive, less durable mirror coatings would be required, and a commercial lens may not perform adequately. The R-band is probably optimal, and a commercial lens can be refocused to perform well at these wavelengths.

Technical Considerations

A CCD that meets the requirements of this design is currently manufactured by SITe (formerly Tektronix), which offers a 1024 x 1024 pixel device with 24 m pixels and dark currents less than 1 e− @ −30 C. Photometrics, Princeton, and other companies offer turn-key cameras using this device, with acceptable read noises (see Appendix 3). All these packages offer subarray readouts, decreasing the readout time without increasing the readout noise.

The mirrors we require are not offered off-the-shelf but can be custom made to our specifications by companies such as International Scientific Products, at a cost far less than lenses performing the same function. Re-imaging lenses to capture the collimated beam and give the requisite f/2.5 imaging onto 24 m pixels are commonly available from any major camera company like Nikon or Canon. For R-band guiding, these lenses have an "infrared" refocus dial.

Optical Design

All guider optical Zemax design files can be found in the altair/c/zmx404/nirspec/guider directory. The final end-to-end design is gcam00.zmx, which is in millimeters. The design expressed in meters is in cdrgcam.zmx, and tolerances were generated in gcam00tl.zmx.

The full optical prescription of the guider system is included in Appendix 4. Table 3 shows the optical and mechanical specifications for the guider mirrors. The substrates should not exceed the dimensions given in Table 3 or they will run into the dewar wall. The surfaces must meet the performace specifications out to within 3 mm of the edge.

TABLE 3. Guider Optical Prescription

Mirror	Radius

(mm)	Major axis (mm)	Minor axis (mm)	Thicknessa (mm)
flat	infinity	233.0	165.0	23.0
(flat hole)	-	83.2	58.8	-
sphere	1060.3	190.0	190.0	-
aThickness refers to the physical thickness of the mirror substrate, not a distance between surfaces. No thickness is listed for the spherical mirror because no restrictions apply to it.

Note that the beam footprint at the flat is elliptical. The dimensions given for this mirror and its hole are for the major and minor axes of these ellipses. The hole must be an ellipse, but the mirror can be rectangular. The edge of the hole must be chamfered at 45E in the long axis, so that neither beam "sees" the substrate. It is preferable, but not necessary, to have the edge of this mirror that is closest to the dewar chamfered at 45E. The thickness of this mirror must not exceed 23 mm, or one tenth of its major axis, to allow sufficient room for the lens behind it. No thickness restrictions apply to the spherical mirror.

Optomechanical tolerances were generated by limiting the degradation of spot size, plate scale, and clearance through the hole in the flat mirror and entrance aperture of the commercial lens. Table 4 shows the tolerances that must be met to ensure that these constraints are preserved.

TABLE 4. Guider Optomechanical Tolerances

Tolerance	Units	flat	flat hole	sphere
axes (diam)	mm	± 2	± 0.5	± 2
radius	mm	-	-	+ 9

- 11
surface figure	l632.8nm rms	4	-	2
surface quality	-	60/40	-	60/40
surface roughness	Angstroms	150	-	150
thickness	mm	+ 0	-	-

The tolerance on the hole dimensions are most critical at the lowermost position as viewed in Figure 1, where the collimated beam is at its closest point. A tighter tolerance than that listed above is desirable at that point, whereas the rest of the hole could be kept to a looser tolerance if necessary.

Spot diagrams and geometric encircled energy plots in Figure 2 are traced for three field points: inner, middle, and outer points on the guiding annulus. The end-to-end performance does not include aberrations from the final re-imaging lens, since its design is unavailable to us. By the same token, diffraction encircled energy could not be measured either. The ray-trace design simply uses a perfect "paraxial" lens with the correct focal length in place of the real lens. Because of this, the ray-trace is independent of wavelength. Nevertheless, the spots at all field points are substantially smaller than the size of one pixel. At visible wavelengths diffraction effects will be minimal, and the performance of commercial lenses are usually close to diffraction-limited.

Figure 2. Spot diagrams and geometric encircled energy plots for inner, middle, and outer field points in the guider annulus. Box size in spot diagrams is equal to one pixel. Rightmost edge of encircled energy plot represents the energy captured in one pixel.

Summary

The simple, fixed guider system appears to be able to satisfy all of the defined performance requirements. That is, it is easy to scan the field in search of a guide star since the whole field is imaged at once; it can hold a guide star all night; its field of view provides a low probability of not finding a suitable star; and in theory it can guide to acceptable accuracy anywhere in the sky except for a few minutes at the smallest zenith angles. Optical models show the performance to be more than satisfactory at all field points without the need for a focus mechanism, and all necessary components for this system can be acquired either commercially or custom-made.

APPENDIX 1. Guider Centroid Error with V=20 Star, 0.7" seeing @ full moon

APPENDIX 2. Guide Star Speed (in field arcseconds/second).

APPENDIX 4. Full Optical Prescription for the Guider Camera

System/Prescription Data

File : G:\ZMX45\NIRSPEC\GUIDER\GCAM00.ZMX

Title: NIRSPEC Guider: 85mm f/1.8

Date : SAT MAY 25 1996

GENERAL LENS DATA:

Surfaces : 15

Stop : 2

System Aperture :Entrance Pupil Diameter

Ray aiming : Off

Apodization :Uniform, factor = 0.000000

Eff. Focal Len. : -24229.7 (in air)

Eff. Focal Len. : -24229.7 (in image space)

Total Track : 19255.7

Image Space F/# : 2.42297

Working F/# : 2.4737

Obj. Space N.A. :4.99999e-007

Stop Radius : 5000

Parax. Ima. Hgt.: 12.0269

Parax. Mag. : 0

Entr. Pup. Dia. : 10000

Entr. Pup. Pos. : 16650

Exit Pupil Dia. : 68.4762

Exit Pupil Pos. : -165.958

Field Type : Angle in degrees

Maximum Field : 0.02844

Primary Wave : 0.700000

Lens Units : Millimeters

Angular Mag. : -146.036

Fields : 5

Field Type: Angle in degrees

# X-Value Y-Value Weight

1 0.000000 -0.012700 1.000000

2 0.000000 -0.021000 1.000000

3 0.000000 -0.028440 1.000000

4 0.000000 0.028440 1.000000

5 0.000000 0.012700 0.000000

Vignetting Factors

# VDX VDY VCX VCY

1 0.000000 0.000000 0.000000 0.000000

2 0.000000 0.000000 0.000000 0.000000

3 0.000000 0.000000 0.000000 0.000000

4 0.000000 0.000000 0.000000 0.000000

5 0.000000 0.000000 0.000000 0.000000

Wavelengths : 1

Units: Microns

# Value Weight

1 0.700000 1.000000

SURFACE DATA SUMMARY:

Surf Type Radius Thickness Glass Diameter Conic

OBJ STANDARD Infinity Infinity 0 0

1 STANDARD Infinity 16650 491.4 0

STO STANDARD -34974 -15394.99 MIRROR 10000 -1.003683

3 STANDARD -4737.916 17818.62 MIRROR 1309.857 -1.644326

4 COORDBRK -------- 0 0 --------

5 STANDARD Infinity 0 222.9301 0

6 STANDARD Infinity 0 MIRROR 222.9301 0

7 COORDBRK -------- -490.8212 0 --------

8 STANDARD 1060.308 490.8212 MIRROR 179.1102 0

9 COORDBRK -------- 0 0 --------

10 STANDARD Infinity 0 81.2 0

11 COORDBRK -------- 74 0 --------

12 STANDARD Infinity 11.4325 42.42345 0

13 STANDARD Infinity 8.7325 45.8 0

14 PARAXIAL -------- 87.85641 48.37912 --------

IMA STANDARD Infinity 0 24.21961 0

SURFACE DATA DETAIL:

Surface OBJ : STANDARD

Surface 1 : STANDARD

Aperture : Circular Obscuration

Minimum Radius : 0

Maximum Radius : 654.929

Surface STO : STANDARD

Aperture : Circular Aperture

Minimum Radius : 654.929

Maximum Radius : 1e+010

Surface 3 : STANDARD

Surface 4 : COORDBRK

Decenter X : 0

Decenter Y : 0

Tilt About X : 45

Tilt About Y : 0

Tilt About Z : 0

Surface 5 : STANDARD

Comment : PICKOFF HOLE

Aperture : Elliptical Obscuration

X Half Width : 29.5

Y Half Width : 41.6

Surface 6 : STANDARD

Comment : PICKOFF FLAT

Aperture : Elliptical Aperture

X Half Width : 82.4

Y Half Width : 116.5

Surface 7 : COORDBRK

Decenter X : 0

Decenter Y : 0

Tilt About X : 45

Tilt About Y : 0

Tilt About Z : 0

Surface 8 : STANDARD

Comment : COLLIMATOR

Aperture : Circular Aperture

Minimum Radius : 0

Maximum Radius : 95

Surface 9 : COORDBRK

Decenter X : 0

Decenter Y : 0

Tilt About X : -45

Tilt About Y : 0

Tilt About Z : 0

Surface 10 : STANDARD

Comment : PICKOFF HOLE

Aperture : Elliptical Aperture

X Half Width : 29.5

Y Half Width : 41.6

Surface 11 : COORDBRK

Decenter X : 0

Decenter Y : 0

Tilt About X : 45

Tilt About Y : 0

Tilt About Z : 0

Surface 12 : STANDARD

Comment : FRONT OF LENS

Aperture : Circular Aperture

Minimum Radius : 0

Maximum Radius : 35.5

Surface 13 : STANDARD

Comment : 1ST LENS ELEMENT

Aperture : Circular Aperture

Minimum Radius : 0

Maximum Radius : 23.6

Surface 14 : PARAXIAL

Comment : PRINCIPLE PLANE

Focal length : 84.815

Surface IMA : STANDARD

Comment : CCD

Aperture : Rectangular Aperture

X Half Width : 12.3

Y Half Width : 12.3

SOLVE AND VARIABLE DATA:

Semi Diam 1 : Fixed

Semi Diam 2 : Fixed

Semi Diam 3 : Fixed

Thickness of 7 : Variable

Parameter 3 Surf 7: Pickup from 4 times 1.000000

Curvature of 8 : Variable

Thickness of 8 : Solve, pick up value from 7, scaled by -1.00000

Parameter 3 Surf 11: Pickup from 9 times -1.000000

Thickness of 14 : Variable

INDEX OF REFRACTION DATA:

Surf Glass 0.700000

0 1.00000000

1 1.00000000

2 MIRROR 1.00000000

3 MIRROR 1.00000000

4 1.00000000

5 1.00000000

6 MIRROR 1.00000000

7 1.00000000

8 MIRROR 1.00000000

9 1.00000000

10 1.00000000

11 1.00000000

12 1.00000000

13 1.00000000

14 1.00000000

15 1.00000000

F/# DATA:

Wavelength: 0.700000

# Field Tan Sag

1 0.0000, -0.0127 deg: 2.4719 2.4738

2 0.0000, -0.0210 deg: 2.4686 2.4738

3 0.0000, -0.0284 deg: 2.4641 2.4737

4 0.0000, 0.0284 deg: 2.4641 2.4737

5 0.0000, 0.0127 deg: 2.4719 2.4738

GLOBAL VERTEX COORDINATES AND DIRECTIONS:

Surf X coord Y coord Z coord X direc Y direc Z direc

1 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000

2 0.000000 0.000000 16650.000000 0.000000 0.000000 1.000000

3 0.000000 0.000000 1255.015000 0.000000 0.000000 1.000000

4 0.000000 0.000000 19073.633360 0.000000 -0.707107 0.707107

5 0.000000 0.000000 19073.633360 0.000000 -0.707107 0.707107

6 0.000000 0.000000 19073.633360 0.000000 -0.707107 0.707107

7 0.000000 0.000000 19073.633360 0.000000 -1.000000 -0.000000

8 0.000000 490.821158 19073.633360 0.000000 -1.000000 -0.000000

9 0.000000 0.000000 19073.633360 0.000000 -0.707107 0.707107

10 0.000000 0.000000 19073.633360 0.000000 -0.707107 0.707107

11 0.000000 0.000000 19073.633360 0.000000 -1.000000 -0.000000

12 0.000000 -74.000000 19073.633360 0.000000 -1.000000 -0.000000

13 0.000000 -85.432500 19073.633360 0.000000 -1.000000 -0.000000

14 0.000000 -94.165000 19073.633360 0.000000 -1.000000 -0.000000

15 0.000000 -182.021409 19073.633360 0.000000 -1.000000 -0.000000

ELEMENT VOLUME DATA:

Units are cubic cm.

Values are only accurate for plane and spherical surfaces.