Embed Size (px)
Citation preview
Analytical Study of the Large
Orbital X-Ray Telescope Imaging System
Contract No. NAS8-29855
FINAL REPORT .
.by
J.- William Foreman, Jr.
Joseph M. Cardone
August, 1973
(NASA-CR-124429) ANALYTICAL STUDY OF THE N73-32338
LARGE ORBITAL X-RAY TELESCOPE IMAGINGSYSTEM Final Report (Montevallo Univ.,Ala.) 46 p HC $4.50 CSCL 20F Unclas
G3/14 15621
Submitted to
National Aeronautics and Space Administration*,
George C. Marshall Space Flight Center -
Marshall Space Flight Center, Alabama .. ,
by
UNIVERSITY OF MONTEVALLO
Montevallo, Alabama ...
.5 . ' ., , ..
https://ntrs.nasa.gov/search.jsp?R=19730023606 2020-03-14T19:30:47+00:00Z
TABLE OF CONTENTS
Page.
I. INTRODUCTION ...... . .......... 1
II. PROGRAM OUTLINE................ ........ . 2
III. RESULTS............................ . . ............. . . 5
Table: Maximum and rms spot diameters in thefocal plane as a function of off-axisangle for a point source at infinity......... 6
Table: Location of centers of spots formed bythe individual mirrors as a function ofoff-axis angle, showing misregistrationof the spots .......................... . .... * 7
Representative spot diagrams for a point sourceat infinity....... ............................ 8
Table: Maximum and rms spot diameters in theplane of best on-axis focus as a functionof off-axis angle for a point source ata finite distance of 1000 feet fromthe LOXT..................................... 13
Table: Plane of best focus as a function ofoff-axis angle for a point sourceat infinity ....... ........................ 14
Graph: Surfaces of best focus for a pointsource at infinity................... .... .. 15
Graph: Reflectivityrersus glancing angleof incidence for various x-ray wave-lengths ................................. 16
Representative line spread functions for thecomposite system for a source at infinity......... 17
Representative line spread functions for thecomposite system for a source at a finitedistance of .1000 feet from the LOXT................. 31
IV. REFERENCES"...44
I. INTRODUCTION
This report presents the results of an analytical study of the
Large Orbital X-Ray Telescope (LOXT) designed'by American Science and
Engineering. The LOXT consists of a nested array of four conventional
paraboloidal-hyperboloidal x-ray telescopes arranged with a common
optical axis and a common focal plane. The composite nested array has
a nominal effective focal length of 135.0 inches. The equations of
the various mirror surfaces'and the numerical values of the parameters
in the defining equations are given in the technical specification
sheets for the LOXT issued by American Science and Engineering.
The work reported herein was carried out by two faculty members
of the Department of Mathematics and Physics at the University of
Montevallo: Dr. J. William Foreman, Jr. and Mr. Joseph M. Cardone.
Dr. Foreman served as Principal Investigator for the study. This
report covers work done during the entire contract period from
1 July, 1973.to 31 August, 1973.:
Except for computer graphics, all computer programs were run in .
the automatic extended precision mode on an XDS-Sigma 5 computer in
Wing C of the Astrionics Laboratory at NASA-MSFC. Spot diagrams and
line spread functions were plotted by a Calcomp Model 566 digital
plotter controlled by an IBM-1130 computer located in Wing B of the
Astrionics Laboratory.
The results reported herein were obtained by the general ray
trace methods documented in Ref. 1. A more detailed treatment of the
analytical approach will be given iA the final report for Contract
NAS8-30375, which is a continuation of -the work begun in the present
contract.
. .. 1
II. PROGRAM OUTLINE
The analytical study of the LOXT consisted of five distinct parts:
(1) Calculation of the rms and maximum spot diameters in the focal.
plane as a function of off-axis angle for a point source at infinity,
and plotting of representative spot diagrams. These data were obtained
for each of the four paraboloidal-hyperboloidal mirror sets individually,
and also for the composite system (all four mirrors operating simul-
taneously).
(2) Calculation of the rms and maximum spot diameters in the plane of
best on-axis focus as a function of off-axis angle for a point source
at a finite distance of 1000 feet from the LOXT. Data were obtained
for each mirror set individually, and.for the composite system.
(3) Determination of the field curvature of each paraboloidal-hyperboloidal
mirror set, and the field curvature of the composite system, for a point
source at infinity.
(4) Calculation of the radial and tangential line spread functions
for the composite system in the focal plane at various off-axis angles
for a line source at infinity.
(5) Calculation of the radial and tangential line spread functions
for the composite system in the plane of best on-axis focus at various
off-axis angles for a line source at a finite distance of 1000 feet
from the LOXT.
NOTE: In Parts (4) and (5), the. line spread functions were
calculated at two'different x-ray wavelengths, 4 and 40 R. The
line spread functions were also calculated assuming 100% reflection
at every mirror surface, for comparison with the 4 and 40 2 results.at --
In the ray trace analysis of the LOXT, the origin of the coordi':
nate system was chosen to lie on the optical axis-in the focal plane of
the telescope, as shown in Fig. 1. The x-axis was chosen to be the
optical axis, to conform with the coordinate system used by American
Science" and Engineering in their specification sheets.
-.3
-. .- . .
.. . /. ". .cene -,'. Enieri'~n .hi .pcfto-he .; ..
. . -. .' .
S... •: _ , .. .: .
.. . ::
• . ., . [ : . ."i .•. - i• , . . .
y
FOCAL PLANE
HYPERBOLOIDS PARABOLOIDS
OPTICAL AXIS
I -S*] . .. '" * I
I *** .* *.,".... . I *: - .. ...
I
. 135.0 in. .
iP-i
Figure 1. Coordinate system used in the ray trace analysisof.the LOXT.
III. RESULTS:
In this section, the quantitative results-of the analytical
study of the LOXT are presented 'in tables and graphs, each of which
is self explanatory..
5
Maximum and rms spot diameters in thefocal plane as a function of,off-axisangle for a point source at infinity.
NOTE: 190 rays were traced through each individual mirror to obtain thespot sizes listed in the table below. 760 rays, 190 through eachmirror, were used to obtain the tabulated spot sizes for thecomposite mirror system.
OFF-AXIS Mirror RMS Spot Maximum SpotANGLE No. Diameter Diameter
(arc-minutes) (arc-seconds) (arc-seconds)
1 0.112147 0.178788
2 0.090454 0.160892
0.5.3 0.079468 . 0.152659
4 0,080983 0.155859
Comp. 0.120287 0.290426
1 0.267863 0.496689
2 0.249443 0.483829
1.0 3 0.259901 0.498130
4 0,299929 0.548155
Comp. 0.312344 0.721699
1 0.491949 0.952448
2 0.500798 0.967121
1.5 3 0.557542 1.034049
4 0.665093 1.172775
Comp. 0.,607077 1.293320
1 0.794427 1.544891
2 0.849777 1.609202
2.0 3 0.974122- 1.758263
4 1.177061 2.026821
Comp. 1;.013435 2.2187026
• ,: .
Location of centers of spots formed by the individualmirrors as a function of off-axis angle, showingmisregistration of the spots.
OFF-AXIS ANGLE MIRROR DISPLACEMENT OF SPOT MISREGISTRATION* .(arc-minutes) NO. CENTER FROM ORIGIN (arc-seconds) ,
(inches)
1 . 0.019737
2 0.0197090.5 0.104
3 0.019687
4 0.019669
1 . 0.039474
2 0.0394181.0 0.211
3 0.039373
4 0.039336
' 1 0.059209
2 0.0591251.5 0.318
3 0.059056
4. 0.059001
1 0.078943
2 0.0788302.0 0.432
3 0.078737
4 '0.078660
Misregistration is calculated as SPOT DISPLACEMENT(Mirror #1), - SPOTDISPLACEMENT(Mirror #4), expressed in arc-seconds instead of inches.
ii .... ; .~ ~ :::/ .. '' ,, , : 1 :7 L i.. "
OFF-AXIS ANLE = . O ARC-MINUTES
-. ..
MIRROR. SYSTEM ND i.
POINT SCURCE AT INFINITY
-NE LE IVIND
A8. ..
N L V=OA8
±x1
t
x -
C'FF-AXI53 ANDLE' "I -.0 ARC-MINUJTES
'MURRuR 'SYSEM NO- 2.
PO-,I NT £OLjRCE AT INFINITY
ONtE scALrL OTVi~3ic!N Q' AR-E-LOND
9
X X IN X1
x
EFAXI ANW x x~ Am-INTE
MIRF~CFK SYTE \'X" 3
xUN xCfC AT INFIT
'101
x xxxx
I X X~~ X xx 'x xXXX
X xxx XxX xxxxxx
11 *,* . X
J''
x 4 x
~2xx4x'xx
x t
*OF-XIx- ANL x F~-IU
CCMOETT M~fOF 9~3EJ
PON 3FC TIFNT
ONESfAL: DVI3TN >1 ~C9EON
x 12
Maximum and rms spot diameters in the plane ofbest on-axis focus as a function of off-axis'angle for a point source at a finite distanceof 1000 feet from the LOXT. The plane of beston-axis focus is x = -1.53429 inches for thecomposite mirror system.
NOTE: 190 rays were traced through each 'individual mirror to get thespot sizes listed below. .760 rays were traced to obtain the datafor the composite mirror system.
OFF-AXIS MIRROR RMS SPOT MAXIMUM SPOTANGLE NO. DIAMETER DIAMETER
(arc-minutes) (arc-seconds) (arc-seconds)
1 1.137369 2.042655
2 0.791560 1.317691
0.0 . 3 0.795116' 1.524142
4 0.821337 1.509541
Comp. 0.898190 2,042655
1 1.160135 2.307675
2 0.867065 2.054933
0.5 3 0.904466 2.303139
4 0.954059 2,333116
Comp. 0.981017 2.428256
1 1.237883 2.545708
2 1.080768 2.986181
1.0 3 1.197364 3.314195
.4 1.305846 ' 3.442835
Comp. 1.218110 3.635140
1 1.393410 3.397412
2 1.413203 4.114219
1.5 3 1.568367 4.477640
4 1.704656 4.397936
Comp. 1.537094 4.583448
..... - ':>: '1~: 1 3 " ' " .
OFF-AXIS MIRROR RMS SPOT X-Coordinate ofANGLE NO. DIAMETER PLANE OF BEST
(arc-minutes) (arc-seconds) FOCUS._(inches)
1 0.0524019 0.00003
2" 0.0385480 0.00005
0.25 3' 0.0276298 0.00007
4 0.0197450 0.00011
Comp. 0.0537278 0.00005
1 0.106515 0.00014
2 0.0799889 0.00019
0.50 3 0.0605709 0.00028 X
14.4
4 0.0495623 0.00043
Comp. 0.111681 0.00022 0o
1 0.163495 0.00032 .r
2 0.126384 0.00045
0.75 3 .0.102226 0.00064
4 0.0938131 0.00098 o
Comp. 0.177111 0.00050
1 0.224393 0.00058
2 0.179394 0.00080 44 0Oa
1.00 3 0.154674 0.00115 w
4 0.153955 0.00175 (
Comp. 0.252615 0.00090
1 . 0.290131 0.00091
2 _ " _ 0.240288 0.00125
1.25 3 0.219079 ; 0.00180
4 0.231553 .- .00273
Comp. 0.340191. 0.00142
1 0.361512 0.00131
2 0.309976 0.00181
1.50 1.50 0.296068 0.00260
4 0.325193 0.00394
Comp. 0.441103 0.00205
14 " " , ,
0.06 1 2 Composite 3 4
0.05
0.04 -
0.03 -
0.02
0.01
0.00
0.00 . 0.001 0.002 . 0.003 0.004
X (inches)
Surfaces of best focus -for a point source at infinity.
1.0
40
- 0.5 d
0.o-.
0 20 40 60 80 100 120 140 160 180
GLANCING ANGLE OF INCIDENCE (Arc-minutes)
Reflectivity versus glancing angle of incidence for various x-ray wavelengths.
(Numerical data for these curves was taken from the reflectivity curves fornickel given in Fig. 2-2, p. 2-6 of Reference 2.)
LINE SPREAD FUNCTIONS - COMPOSITE SYSTEM - SOURCE AT INFINITY
The line spread functions shown below.were computed by the general method
outlined in Ref. 1.. A square grid with 25 columns and 25 rows was used
to collect the rays in each case. Thus, each line spread function is
plotted as a histogram 25 bars in length.
The source at infinity was taken in every case to lie in the xy-plane.
Thus, the resulting spots are centered on the y-axis and are symmetrical
about the xy-plane.. Consequently, all line spread functions L(z) are
symmetrical about their centers. Approximately 5000 rays were traced
to obtain the line spread functions at each off-axis angle.
The line spread functions were calculated for , = 40 and h = 4
by using the reflectivity curves shown in the graph on the previous
page. The contribution from each ray was taken to be R (0 )R (9 ),Pp h h
where R is the reflectivity for the glancing angle 9 at the parabolicP .p
mirror and Rh is the reflectivity for the glancing angle 9h at the
corresponding hyperbolic mirror. The line spread 'functions were also
calculated assuming uniform reflectivity C R 1.0 ) for comparison
with the results at 40 and 4 :
17
z z
25 columns
point source
(in xy-plane)
The line spread function L(y) is obtained by summing contributionsfrom all rays across each row.
The line spread function L(z) is obtained by summing contributions.from all rays along each column.
All line spread functions are normalized to unity at their maxima.The units on the vertical axes in the line spread histograms whichfollow are thus 0.1, 0.2, "', 1.0. Each scale division on thehorizontal axes represents 0.1 arc-second.
18
--- --
19
I i* .I I i !: I I ! ! ,
COMPOSITE MIRROR SYSTEM - SOURCE AT' INFINITY
ONE SCALE DIVISION = 0,, ARC-SECOND " . "
!" - -v- - -" - -- --: -- " -
LINE SPREAD FUNCTION L (Z
AVELENGT ANGSTROMS .
O FF-AXIS ANGLE = 0c5 ARC-MINUTES "
LINE SPREA FUNCTION L(Z).
COMPOSITE.MIRROR SYSTEM - :SOURCE AT INFINITY
II ONE SCALE DIVISION O"i A:C-SECONO
WAVELENGTH - 4 ANSTRnMS-. ' ' " ' "" ' '"'
:2 0
I I I I ... Ii .1 1 1 1 1 1 1 1
tti
OFF-AXIS ANGLE =' 05 ARC-MINUTES
LINE SPREAD FUNCTION L Y)
MT
COMPOSITE MIRROR SYSTEM - SOURCE AT INFI TTY
ONE SCALE DIVISION. O-i ARC-SECONO
-WAVELENGTH = 40, ANGSTROMS:
OFF-AXIS ANGLE = O5 ARC-MINUTES
LINE SPREAD FUNCTION L (Z)
COMPOSITE MIRROR SYSTEM - SOURCE AT INFINITY -
ONE SCALE DIVISION = Oi ARC-SECOND
: WAVELENGTH !' 40 A'NGSTROMS
22
a i
%i
NE SCALE I-ION 0 ARC-
iIVITY (
UFORM RELEC .= ,A
LINE SPRED FU~NCTION L[ :
COlviDOSITE MvIRROR SYSTEM - SO RIIC~E AT INFINITY
ONE :SCAL DIVISION = O,,1 RC-SECOND
UNIFORM REFLETIVITY. CRji,,O)
..; ': " .2 3i : -: "': ' " ' - .
24
.. .. . -
OFF-AXIS' ANGE = O5 ARC-MINUTES
UNIFORM REFEETIvITY [R=iaO)
?~:: . :r :: ' : :: A
241 i.
? I I .- i 1,-
F 4 ' i ,. • ." , .,
OFF-AXIS ANGLE = 1 5 ARC MINUTES
LINE SPREAD FUNCTIQN L (Y
COMPOSITE MIRROR SYSTEM - SOURCE AT INFINITY
ONE SCALE DIVISION .0 i ARC-SECOND
WAVELENGTH 4 ANGSTROMS
ti
I -
OFF-mXIS ANGLE = 1.' ARC-MINUTES:
LINE SPREAD FUNCTION L(Z)
COMPOSITE MIRROR- SYSTEM - SOURCE AT INFINITY
ONE SCALE DIVISION = 0± ARC-SECONO
WAVELENGTH= '4 ANGSTROMS
: :,26'
LINE SPREAD FUNCTION L(Y)
COMPOSITE MIRROR SYSTEM - SOURCE AT INFINITY
ONE .SCALE, VISION Oi ARC-SE OND
FWAVELENGTH = 40 ANRGS-iTRMS
I. -I . . : 1.
OFF-AXIS ANGLE -= 5 ARC-MINUTES
LINE ,SPREAD FUNCTION L (Z).
COMPOSITE MIRROR SYSTEM -SOURCE AT INFINITY
ONEt SCALE DIVISION = Oci ARC-SELONO
WAVELENGTH = 40 ANGSTROMS
28
IL
OFF-AXIS ANGLE = i 5 ARC-MINUTES
LINE SPREAD FUNCTION L(Y)
COMPOSITE MIRROR SYSTEI4 - SOURCE AT INFINITY
ONE SCALE DIVISION = O1 ARC-SECOND
UNIFORM REFLECTIVITY (R=±O).
t3i
LINE SPREAD FUNCTION L(Z)
COMPOSITE MIRROR SYSTEM - SOURCE AT INFINITY
ONE SCALE DIVISION = 0° lARC-SECONO i
UNIFORM REFLECTIITY:'.(R=:O: '
30 : ".
LINE SPREAD FUNCTIONS - COMPOSITE SYSTEM - FINITE SOURCE DISTANCE
The general procedure for calculating the line spread functions
for a finite source distance of 1000 feet is the same as that
used for a source at infinity.
- ~ : :; ., j • . .
~~i : :'
:::: i
+L
W N = 4 . NSRM
I I Iii IT I I IlI I
OFF-AXIS AN lE = O;O ARC-MINUTES
LINE SPREAD FUNCTION L(Y)
COMPOSITE MIRROR SYSTEM - SOURCE AT FINITE DISTANCE
ONE SCALE DIVISION =Q'01 ARC-SECOND
WAVELENGTH = 4 ANGSTROCMS
.: ': I32 ,•
OFF-AXIS ANGLE = 0O0 ARC-MINUTES
LINE SPREAD FUNCTION L(Z)
COMPOSITE MIRROR SYSTEM - SOJRC: AT FINITE ISTANCE
WAVELENGTH = 4 ANSTROMSI.
33
. . • ' . ' : , . /
.. .
ill
III
. ..
II~~ I I I .i I IIII
OFF-AXIS ANGLE = OO ARE.-MINUTES
LINE SPREAD FUNCTION L(Y)
COMPOSITE MIRROR SYSTEM - SOURCE AT EINITE DISTANCE
ONE SCALE-OIVISION.= 0-i ARC-SECOND
SWAVELENGTH 4O ANGSTROMSS
34 ,. .- . . .34
_ : , . % ,- ,, ,
+ I.... .I . . . . . . I I I I I I
OFF-AXIS ANGLE = O.O ARC-MIT NUTES
LINE SPREAD FUNCTION L (Z)
COMPOSITE MIRROR SYSTEM . SOURCE AT FINITE DISTANCE
ONE SCALE .DIVISION = Oci ARC-SECONO
WAVELENGTH = 40 ANGSTROMS'.
35'
rI
OIFF-AXIS ANGLE .= OrjAlRC:-M INUTES
LINE SPREAD FUNCTION L (Y)
CMPOS.ITE .MIRRR SYSTEM - SOUCE 'AT FINITE DTSTANCE
.O : ii 'A i .
ONE SCALE DIVISION = Oi ARC-SECOND
UNIFORM REFLECTIVITY (R= o) KO
36
I!
OFF-AXIS.ANGLE = OJ -ARC-MINUTES
LINE SPREAD FUNCTION L (Z)
COMPOSITE MIRROR. SYSTEM - SOURCE AT FINITE DISTANCE
ONE SCALE DIVISION = Oi ARC-SECOND
S-UNIFORM REFLECTIVITY (R-iO)'
37
-I l
I S
OFF-AXIS ANGLE = i5 ARC-MINUTES
LINE SPREAD. FUNCTION L,(Y)
COMPOSITE MIRROR SYSTEM SOURCE AT FINITE DISTANCE
ONE SCALE ODIVISION = . ARC-SEONO
WAVELENGTH = 4 ANSSTROMS
38
SH H iii i i !-I[
INC- SPREAD 'FONEION (2
II. .
-1,
.r_ "- f'
I ii i i
i.
SOF-AXIS ANG3LEI .'rR-MIr~fjTE
-INE SPREAD FUlNCTIDN L(Z)
COMPOSITE MIRROR SYSTEM.- SOURCE AT FINITE DISTANCE
ONE SCALE CIVISION. OGi ARC-SECOND".:: '. " . . : : : .: . ! :
WAVELENGTH;'= "4 IANGSTROMS .
..... .39.
-"-: 39.
II
OFF-AXIS ANGLE .'5' ARC-MINUTES
LINE SPREAO FUNCTION L (Y)
COMPOSITE MIRROR SYSTEM - SOURCE AT FINITE DISTANCE
ONE SCALE OIVISION = Oi ARC-SECOND
WAVELENGTH = 40 ANGSTROMS
40
OFF-AXIS ANCLE . ,5 ARC-MINUTES .
LINE SPREAD FUNCTION L(Z)
COMPOSITE MIRROR :SYSTEM - SOURCE AT FINITE DISTANCE
ONE' SCALE DIVISION = O. ARC-SECOND
WAVELENGTH -= .40 ANGSTROMS
41
- t-.
I.+
C]LF-AXIS ANLE. =i'- AFIC-MINUTES
LINE SPR~EAD3 FNCTION M~Y
CCMPOSITE MIRROR SYSTEM -SO~URCE AT' FINITE. OISTAND7E
OESCL OIVI9IDN OCi ARC-SECONO.
UNIFIM REFLEL TIV iY ( LO
<tI
Ifi
1 4 441 i i 4t
Si i
ITI
* *L
4-
n jL i 4f-ii :t tij
F-FA){I5 ANGLE 2o5 AiE -MiNUTES
*LINE SPREAfl FUNCTION L(Z)
CMPOSITE MIRRO~ S5YSTEM4 - SOORECE AT FINITE OISTANC:
*ONE S~CALEODIV11S1N = Od± ARL-9EmZND
UNIFOR~M RFLELTIVITY (Rzi.oC)3: . "
i I J
- ' .". ', - '"
REFERENCES
1. J.W. Foreman, Jr., et al., Analytical Study of the Imaging
Characteristics of the Goddard ATM X-Ray Telescope, Report
No. SP-505-0279, Space Support Division, Sperry Rand Corp.,
Huntsville, Alabama (September, 1969).
2. L. VanSpeybroeck and R. Giacconi, Final Report: LOXT Mirror
Design Study, Report No. ASE-3096, American Science and
Engineering, Inc., Cambridge, Massachusetts (October, 1972).
' 'LI
