Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hea-www.harvard.edu/~jcm/asc/docs/coords/coords5.ps Äàòà èçìåíåíèÿ: Mon Oct 18 22:23:19 2010 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 10:53:15 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: angular size

ASC Coordinates: Rev 5.0
SDS­2.0
Jonathan McDowell
1999 Apr 4
Contents
I Part 1: Mission­independent formulation 6
1 General Introduction 6
1.1 Update notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Pixel convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Vector and coordinate notation . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Rotation and translation of Cartesian systems . . . . . . . . . . . . . . . . . 7
1.2.4 Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Data Analysis Coordinate Systems ­ Imaging 8
2.1 General model of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Data Analysis 1: Telemetry to Tangent Plane . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Telemetry to Chip coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Tiled Detector Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3 Local Science Instrument coordinates . . . . . . . . . . . . . . . . . . . . . . 11
2.2.4 Mirror coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.5 Pixel planes for intermediate systems . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Data Analysis 2: Tangent Plane to Sky . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Sky Pixel Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Physical Tangent Plane coordinates . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3 Physical Sky Plane coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.4 J2000 Celestial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Simulation: Sky to Telemetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1

2.4.1 Mirror Spherical Coordinates to Focal Surface Coordinates . . . . . . . . . . 21
2.4.2 Focal Surface or Mirror Nodal Coordinates to CPC coordinates . . . . . . . 21
2.4.3 Chip coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Summary of coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Data Analysis 3: Full treatment with misalignments . . . . . . . . . . . . . . . . . . 24
3 Data Analysis Coordinate Systems ­ Gratings 25
3.1 Data Analysis: Grating data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Grating Zero Order Coordinates (GZO­1.0) . . . . . . . . . . . . . . . . . . 25
3.1.2 Grating Angular Coordinates (GAC­1.0) . . . . . . . . . . . . . . . . . . . . 27
3.1.3 Grating Di#raction Coordinates (GDC­1.0) . . . . . . . . . . . . . . . . . . . 27
3.1.4 Grating Di#raction Plane Pixel Coordinates (GDP­1.1) . . . . . . . . . . . . 27
3.1.5 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
II Part 2: AXAF systems 28
4 ACIS 28
4.1 Instrumental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.1 ACIS readout coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.2 ACIS Fast Window mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 2­D detector coordinates: TDET parameters . . . . . . . . . . . . . . . . . . . . . . 30
4.3 3­D chip locations: CPC to LSI transformation parameters . . . . . . . . . . . . . . 31
5 HRC 34
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.1 3­D chip locations: CPC to LSI parameters . . . . . . . . . . . . . . . . . . 34
5.1.2 2­D detector coordinates: TDET parameters . . . . . . . . . . . . . . . . . . 35
5.2 Instrumental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2.1 HRC physical layout and Tap Coordinates (HRC­6.0) . . . . . . . . . . . . . 36
5.2.2 Deriving linear tap coordinates from HRC telemetry . . . . . . . . . . . . . . 36
5.2.3 HRC Chip Coordinates (HRC­1.1) . . . . . . . . . . . . . . . . . . . . . . . 38
6 The SIM 44
6.1 Chip orientation summary tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.2 Relative positions of instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Aimpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2

7 The HRMA (flight) 45
7.1 HRMA nodal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2 Focal and Tangent plane systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2.1 Mosaicing XRCF images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.3 Angular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.3.1 HRMA Left Handed Spherical Coordinates (AXAF­HSC­1.1) . . . . . . . . . 47
7.3.2 HRMA Right Handed Spherical Coordinates (AXAF­HSC­1.2) . . . . . . . . 48
7.3.3 HRMA rotation coordinates (Pitch and Yaw) (AXAF­HSC­3.0) . . . . . . . 48
7.3.4 HRMA Source coordinates (AXAF­HSC­2.1) . . . . . . . . . . . . . . . . . . 49
8 HETG and LETG 49
9 XRCF 51
9.1 XRCF coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
10 ACIS­2C 51
11 HSI 53
11.1 Instrument origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.2 HRMA motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.2.1 XRCF to MNC transformation . . . . . . . . . . . . . . . . . . . . . . . . . 55
11.3 HXDS Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.4 FAM Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.5 FAM Motion, old copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
12 AXAF Spacecraft 58
12.1 Project coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
12.1.1 Spacecraft coordinates (SC­1.0) . . . . . . . . . . . . . . . . . . . . . . . . . 58
12.1.2 HRMA coordinates (HRMA­2.0) . . . . . . . . . . . . . . . . . . . . . . . . 59
12.2 Project Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
12.2.1 Orbiter coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
12.2.2 Payload coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
12.2.3 The Telescope Ensemble Coordinate System . . . . . . . . . . . . . . . . . . 62
12.2.4 Optical Bench Assembly system . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.2.5 OTG Coordinates (OTG­2.0) . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.2.6 Project FPSI Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 62
12.2.7 SIM and ISIM coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.2.8 SAOSAC coordinates (HRMA­3.1) . . . . . . . . . . . . . . . . . . . . . . . 63
12.2.9 Summary of useful HRMA Cartesian systems . . . . . . . . . . . . . . . . . 63
3

List of Tables
1 Default axis orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Tiled Detector Plane systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Parameters of Tiled Detector Coordinate definitions . . . . . . . . . . . . . . . . . . 31
7 ACIS Chip corner locations in ACIS­I LSI coordinates . . . . . . . . . . . . . . . . 33
9 HRC chip (i.e. grid) corner locations in LSI coordinates . . . . . . . . . . . . . . . . 34
10 Tiled Detector Plane systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
11 Parameters of Tiled Detector Coordinate definitions . . . . . . . . . . . . . . . . . . 35
12 HRC­S1 TELV, v and CHIPY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
13 HRC electronically meaningful coordinate ranges . . . . . . . . . . . . . . . . . . . 39
14 HRC­S boundaries (Revised Jan 99) . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
15 HRC­I boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
16 Euler angles in degrees for CPC to LSI coordinates . . . . . . . . . . . . . . . . . . 44
17 Location of instrument origin on Translation Table . . . . . . . . . . . . . . . . . . 44
18 SIM position o#sets for nominal focus positions . . . . . . . . . . . . . . . . . . . . 45
21 GDC pixel image centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
22 GDP Pixel Sizes (assuming flight Rowland radius) . . . . . . . . . . . . . . . . . . . 50
23 Grating properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
24 Tiled Detector Plane systems ­ ACIS­2C . . . . . . . . . . . . . . . . . . . . . . . . 51
25 Parameters of Tiled Detector Coordinate definitions . . . . . . . . . . . . . . . . . . 52
26 ACIS­2C chip corner locations in ACIS­2C LSI coordinates . . . . . . . . . . . . . . 53
27 HSI boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
28 Parameters of Tiled Detector Coordinate definitions . . . . . . . . . . . . . . . . . . 53
29 HSI chip corner locations in HSI LSI coordinates . . . . . . . . . . . . . . . . . . . . 54
30 Location of instrument origin on Translation Table, XRCF . . . . . . . . . . . . . . 54
31 Assumed Misalignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
32 Interesting points in spacecraft and HRMA nodal coordinates . . . . . . . . . . . . 59
33 Geometrical layout of mirrors and detectors: Value of MNC X Coord XN . . . . . . 60
34 Mirror radii, mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
List of Figures
1 Pixel convention. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 The relationship between CHIP and Tiled Detector coordinates. . . . . . . . . . . . 11
3 The relationship between CHIP, LSI and STT coordinates. . . . . . . . . . . . . . . 12
4

4 The instrument compartment (dashed line) may be misaligned with the telescope
mirrors. At calibration, this misalignment (highly exaggerated here) may be sig­
nificant and variable as the mirror is tilted with respect to the instruments. The
instrument bench moves with respect to the instrument compartment, as indicated
by the arrow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 The relationship between STT and STF coordinates. . . . . . . . . . . . . . . . . . 13
6 The Tangent Plane Coordinate system describes the incoming rays, while the Focal
Plane Coordinates system describes the angular position of rays emerging from the
mirrors. The rays then intersect one of several inclined detector planes, causing
events whose locations are described in tiled detector coordinates. . . . . . . . . . . 15
7 The di#erent pixel plane coordinate systems. Distorting e#ects are highly exaggerated. 17
8 Imaging the sky in LSI coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
9 Coordinate systems used in data analysis, 1: Imaging data analysis. . . . . . . . . . 21
10 Coordinate systems used in data analysis, 2: Grating data analysis. . . . . . . . . . 22
11 Grating Zero Order coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12 Grating Di#raction coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
13 ACIS readout nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
14 Relationship of HRC­S pixels to the physical instrument. . . . . . . . . . . . . . . . 40
15 HRC­I pixel axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
16 The AXAF SIM Translation Table, showing the flight focal plane instruments to
scale. Distances are in mm. Coordinate system is AXAF­STT­1.0. . . . . . . . . . . 65
17 HRMA Nodal and STF Coordinates showing the on­orbit configuration. . . . . . . . 66
18 HRMA spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
19 HRMA source coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
20 XRCF Coordinates showing the general configuration with HRMA, DFC and LSI
coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
21 XRCF Coordinates showing the general configuration with HRMA and LSI coordi­
nate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
22 Schematic of interesting points in the spacecraft . . . . . . . . . . . . . . . . . . . . 68
5

Part I
Part 1: Mission­independent formulation
1 General Introduction
There are a lot of di#erent coordinate systems used in the AXAF program, mostly intended for use
in constructing and aligning the hardware. This memo is intended to give the ASC SDS group's
current understanding of the relationships between them (mirror metrology systems are not yet
included) and to define many more systems, which are useful for data analysis of observations both
in flight and at XRCF.
1.1 Update notes
Minor update to XRCF focal plane systems FP­4.1, FP­3.1 (1997 Feb 27); General update post
XRCF, especially to HRC values (1997 Dec 21). Corrected HRC TDET system (1998 May 12).
1.2 Notational conventions
Each of the pixel coordinate systems is identified by a label beginning with the string ASC, followed
by a string identifying the system, and ending with a version number. For example, ASC­CHIP­1.0
(loosely, CHIP coordinates) refers to a pixel system defined by the equation immediately following
its bold­face first mention, and satisfying the pixel convention discussed below.
1.2.1 Pixel convention
In all cases where we use discrete pixel numbers, the corresponding real­valued, continuous pixel
coordinates are defined to be equal to the pixel number at the center of the pixel. We further
recommend that for finite detector planes, one corner be designated as the lower left corner, LL.
Then the pixel which has LL as one of its corners (i.e. the lower left pixel) shall be numbered (1,1)
so that its center has coordinates (1.0, 1.0). The coordinates of the LL point itself are (0.5, 0.5).
If the detector is rectangular with sides of length XMAX, YMAX the pixel coordinates then run
from (0.5, 0.5) in the lower left corner (LL) to (XMAX+0.5, YMAX+0.5) in the upper right corner
(UR) while the pixel numbers in each axis run from 1 to XMAX, 1 to YMAX.
1.2.2 Vector and coordinate notation
A bold face symbol e.g. B denotes a point in 3D space. The notation YA (B) denotes the Y
coordinate of point B in the 3­dimensional Cartesian coordinate system A. When we refer to a
point as an argument in this way we usually get lazy and omit the boldface, e.g. YA (B). The
6

LL
UR
(1.0, 1.0 )
(0.5, 0.5)
(XMAX + 0.5, YMAX+0.5)
(XMAX+0.0, YMAX+0.0)
Figure 1: Pixel convention.
notation PA (B) denotes the triple (XA (B), YA (B), ZA (B)), i.e. the coordinates of B in the A
coordinate system. (P is not in boldface since it is not a vector ­ it is in a specific coordinate
system.)
1.2.3 Rotation and translation of Cartesian systems
We define an Euler rotation Rot(#E , # E , #E ) of a Cartesian system X,Y,Z to be the product of three
rotations
Rot(#E , # E , #E ) = Rot 1 (Z, #E )Rot 1 (Y, # E )Rot 1 (Z, #E ) (1)
where the rotations apply to the successively rotated axes from right to left in the usual sense of
matrix multiplication,
The general transformation from a cartesian system A to a system B involves a scaling, a
translation, and a rotation. This may be described by seven numbers: the scale factor KAB (choice
of units), the position vector PB (A0) = (XB (A0), YB (A0), ZB (A0)) of the origin of A in the B
system, and the three Euler angles #E , # E , #E of the rotation R(A,B) from A to B,
R(A, B) = Rot(#E , # E , #E ) (2)
= # cos #E cos # E cos #E - sin #E sin #E sin #E cos # E cos #E + cos #E sin #E - sin # E cos #E
- cos #E cos # E sin #E - sin #E cos #E - sin #E cos # E sin #E + cos #E cos #E sin # E sin #E
cos #E sin # E sin #E sin # E cos # E
#
Then coordinates of a general point G
PA (G) = (XA (G), YA (G), ZA (G)) (3)
may be converted to
PB (G) = (XB (G), YB (G), ZB (G)) (4)
using the formula
PB (G) = R(A, B)KABPA (G) + PB (A0) (5)
7

If
R(A, B) = Rot(#E , # E , #E ) (6)
then
R(B, A) = Rot(# - #E , # E , # - #E ). (7)
Further, if C is related to B by reflection about the X axis, i.e. YC = -YB , ZC = -ZB then
R(A, C) = Rot(#E , # + # E , # - #E ). (8)
1.2.4 Spherical polar coordinates
We also use spherical polar coordinate systems. The WCS paradigm describes general rotations of
a spherical polar coordinate system. We define the native cartesian axes X,Y,Z of a spherical polar
system (r, #, #) by the equation
(X, Y, Z) = rS(#, #) = (r cos # sin #, r sin # sin #, r cos #) (9)
so that the north pole is through the positive Z axis and the azimuth is zero along the positive X
axis and 90 degrees along the positive Y axis. Any other choice of spherical coordinates (r, # # , # # )
may be defined by specifying the Euler rotation matrix which rotates the corresponding native
systems into each other.
In the appendix I derive
# # = cos -1 (cos # E cos # + sin # E sin # cos(# - #E ))
# # = arg (cos # E sin # cos(# - #E ) - sin # E cos #, sin # sin(# - #E )) - #E
(10)
2 Data Analysis Coordinate Systems ­ Imaging
2.1 General model of the system
We model an observatory consisting of several telescopes. Each telescope has an optical system
(loosely, `mirror') which images the sky by converting incoming photon paths to outgoing photon
paths in a well determined way, and an instrument compartment. In the instrument compartment
is an instrument table on which are mounted several instruments. The instruments are fixed with
respect to the instrument table, but the table may move relative to the instrument compartment.
Further, the instrument compartment itself may move relative to the optical system (for AXAF,
this happens only at calibration). Each instrument consists of one or more fixed `chips' or planar
detector surfaces of finite area (we don't imply by this that they are chips in the semiconductor
sense). For AXAF, these chips are all rectangular, although note that the ROSAT PSPC is circular.
Finally, each chip is subdivided into rectangular (usually square) `detector pixels', enumerating the
8

set of distinct locations that can be represented in the instrument telemetry stream. (In the case
of CCDs, these correspond to physical CCD pixels, while in microchannel plate detectors they are
arbitrarily set by the electronics).
The information available to us, the telemetry position, is two dimensional, but to infer the
final two dimensional angular incoming sky direction we must calculate a photon position in three
dimensional space. Every time we want to make an image, we use a two dimensional pixel plane
(possibly losing information in a third dimension). So the three types of coordinate system we will
use are:
. Two dimensional pixel plane
. Two dimensional spherical angular coordinates
. Three dimensional cartesian coordinates.
2.2 Data Analysis 1: Telemetry to Tangent Plane
Once instrumental details are removed, we derive the CHIP pixel coordinate system which records a
location in a 2­dimensional plane pixel surface. The FP (Focal Plane) pixel coordinate system (called
DETX,DETY in the FITS files) gives the locations in a pixelized tangent plane to the telescope
optical axis. The SKY (X,Y) pixel coordinate system gives locations in a pixelized tangent plane to
the nominal RA and Dec of observation. Going from CHIP to FP involves taking out 3­D geometry
of the chips, position of the instrument on the optical bench, motion of the optical bench, boresights,
and plate scale. Going from FP to SKY involves applying the aspect solution. These three pixel
coordinate systems, plus the convenience TDET system which combines the CHIP systems for all
the components of one detector and systems associated with dispersive gratings, are the systems
that users will see in data analysis. The remaining coordinate systems are used internally to make
precise the transformations involved in going from CHIP to FP to SKY.
2.2.1 Telemetry to Chip coordinates
The telemetry coordinates of a photon are a collection (n­tuple) of integers. The formats may be
very di#erent from instrument to instrument; for instance, the HRC telemetry coordinates consist
of two Tap values and six voltages, while the ACIS telemetry coordinates are directly given as
pixel numbers. For each instrument, we provide a rule to convert from telemetry coordinates to
our standard data analysis chip pixel coordinates which run from 1 to XMAX, 1 to YMAX.
They define a logical plane extending from coordinate 0.5 to XMAX+0.5, 0.5 to YMAX+0.5. In
other words, the center of pixel number (1,1) is (1.0, 1.0), and its lower left corner is (0.5, 0.5).
The logical plane may be larger than the actual set of possibly active pixels. For instance, for a
circular detector such as the ROSAT PSPC, we extend this logical chip pixel coordinate plane to
be rectangular. This chip pixel coordinate system is labelled as ASC­CHIP­1.0.
9

The corresponding physical coordinate system is a three dimensional Cartesian system called
Chip Physical Coordinates (CPC), giving the physical location of a detected photon event on
the active area surface. They are fully defined in terms of the chip pixel coordinates when the pixel
size # p and the array size XMAX x YMAX is given.
(XCPC , YCPC , ZCPC ) are defined to have units of mm. The CPC X and Y axes are coincident
with the chip X and Y axes, and the Z axis completes a right handed set. The CPC Z coordinate
of any point in the chip has a value of 0.0. The X and Y coordinates run from 0.0 to XLEN and
YLEN, where XLEN = XMAX # # p and Y LEN = Y MAX # # p .
Thus if a photon lands at Chip Physical Coordinates XCPC , YCPC its chip pixel coordinates
(ASC­CHIP­1.0) are
CHIPX = XCPC /# p + 0.5 (11)
CHIPY = YCPC /# p + 0.5
or
XCPC = (CHIPX - 0.5)# p (12)
YCPC = (CHIPY - 0.5)# p
Note that CHIPX and CHIPY are by definition linear, by which I mean that the mapping to real
physical space is linear. Now for ACIS it so happens that the true CCD pixels satisfy this condition
su#ciently accurately, but for HRC the readout values may require linearization and removal of
discontinuities.
2.2.2 Tiled Detector Coordinates
In an instrument with multiple detector planes, the planes may be tilted with respect to each other,
or may be separated by a non­integral number of detector pixels, or both (as in ACIS­I). Projecting
onto a 2­D plane (e.g. FP coords) will then lose the identity of individual pixels since one true
detector pixel will map to a variable number of square pixels on the projected plane. For recording
calibration information like bad pixel lists and inspecting the raw image, it is useful to have a single
coordinate system covering the whole instrument which retains true detector pixel identity at the
expense of relative positional accuracy. For this purpose we introduced the tiled detector coordinate
systems (TDET).
# TDETX
TDETY
# = # i
# 1 0
0 H i
## cos # i sin # i
- sin # i cos # i
## CHIPX - 0.5
CHIPY - 0.5
# + # X0 i + 0.5
Y 0 i + 0.5
# (13)
where the values of H i , # i and # i are di#erent for each chip. H i gives the handedness of the planar
rotation and has values +1 or -1, # i gives the sub­pixel resolution factor, and # i gives the rotation
angle of the chip axes with respect to the detector coordinate axes.
10

YCHIP
S
X LSI
LSI
Y CPC
XCHIP, X
CPC
Z CPC
XDET
YDET
XCHIP
YCHIP
Z
LSI
Y
Figure 2: The relationship between CHIP and Tiled Detector coordinates.
2.2.3 Local Science Instrument coordinates
Each instrument has an Instrument Origin which is the nominal focal point for the instrument. For
telescopes with movable instrument tables, the actual focal point for a particular observation may
be di#erent.
To describe the motion of the instrument table, we use three aligned Cartesian coordinate
systems.
. The Science Instrument Translation Frame (STF) coordinate system is fixed in the instrument
compartment, and is used to describe the changing position of the instrument table. Its origin
is at the flight focus and its axes have +X running from the focus toward the telescope
aperture, and +Y and +Z forming a right handed set whose orientation matches that of the
observatory coordinates. For AXAF, the direction of instrument table motion is along Z.
. The Science Instrument Translation Table (STT) coordinate system is fixed in the moving
instrument table, and is used to describe the positions of the instrument origins relative to
each other. The position of its origin is mission­dependent, but the axes are parallel to the
STF axes.
. The Local Science Instrument (LSI) coordinate system for each instrument is fixed in that
instrument, and is identical to the STT frame but with the origin shifted to the instrument
origin.
When the instrument table is moved to put the instrument at its nominal focus position, that
means that the LSI origin is coincident with the STF origin. Since the axes are also parallel,
STF and LSI coordinates become identical. The STF system therefore measures how much the
instrument is o#set from its nominal focus.
The STF, LSI and STT coordinate systems are defined to use units of mm. To convert from
the LSI system to the STT system, one needs to know the STT coordinates of each LSI origin (i.e.
11

the location of each instrument on the instrument table). To convert to the STF system, one needs
to know the instantaneous position of the instrument table, which we describe by giving the STF
coordinates of the STT origin O STT .
To convert from a position P(LSI) in LSI coords to P(STF) in STF coords, one then performs
the vector sum:
P (STF ) = P (LSI) +O LSI (STT ) +O STT (STF ) (14)
adding the STT coordinates of the LSI origin and the STF coordinates of the STT origin. When
the instrument is at its nominal focus, these two vectors are equal and opposite.
To convert from CPC coordinates to LSI coordinates, we need to carry out a rotation and
translation. Each instrument has one LSI coordinate system, and several chips each with its own
CPC system. The information we need for each chip to define the transformation is the LSI
coordinates of each of the four corners of the chip plane (actually, only three of the four are required).
A general point on the plane is
r = p 0 +XCPC eX + YCPC e Y (15)
where p 0 is the origin of CPC coordinates, and eX and e Y are the unit vectors along the CPC axes,
with e Z as the unit normal to the plane.
Let us denote the position vectors of the four chip corners as LL,UL,UR,LR. Then the unit
vectors of the CPC origin and axes in LSI coordinates are
p 0 = LL, eX = LR-LL, e Y = UL-LL (16)
and
e Z = eX # e Y . (17)
YCHIP
S
X LSI
LSI
X
S
Y CPC
XCHIP, X
CPC
Z CPC
Z
Y
LSI
Y
Z
STT
STT
STT
Figure 3: The relationship between CHIP, LSI and STT coordinates.
We can recast this in a rotational formulation,
PG (LSI) = P LL (LSI) +R(CPC,LSI)PG (CPC) (18)
where the matrix is
R(CPC,LSI) =
# # #
(e X ) X (e Y ) X (e Z ) X
(e X ) Y (e Y ) Y (e Z ) Y
(e X ) Z (e Y ) Z (e Z ) Z
# # # (19)
These unit axis vectors can easily be derived from the corner coordinates.
12

Optical bench
Instrument module
platform
SI
Science instrument (described by LSI coords)
Optical bench (described by STT coords)
Instrument module (described by STF coords)
Mirror (described by HNC coords)
Tilt of HNC with respect to STF is exaggerated
Mirror
Figure 4: The instrument compartment (dashed line) may be misaligned with the telescope mirrors.
At calibration, this misalignment (highly exaggerated here) may be significant and variable as the
mirror is tilted with respect to the instruments. The instrument bench moves with respect to the
instrument compartment, as indicated by the arrow.
X STF
S
S
0
HRMA
STF
Z
ACIS Y STF HRC
SIM Translation Table
STT
X
Y STT
STT
Z
Figure 5: The relationship between STT and STF coordinates, AXAF example. The instrument
(SIM) table has moved so that HRC is at the focus.
13

2.2.4 Mirror coordinates
We now specialize to a simple optical system which can be modelled to first order as a thin lens with
an optical node N from which rays appear to emerge toward a detector. This simple description
is fine for the AXAF HRMA, but won't apply directly to the complicated optical path of the
AXAF Aspect Camera. The origin of mirror coordinates (labelled here as MNC for Mirror Nodal
Coordinates, in previous versions of the document HNC for HRMA Nodal Coordinates) is the
nominal optical node of the mirrors. The +X axis goes from the node toward the entrance aperture,
and the Y and Z axes complete a right handed system. To first order, an incoming ray with MNC
direction cosines (-XN , YN , ZN ) emerges from the mirror with direction cosines (-XN , -YN , -ZN ).
(We use the subscript 'N' for mirror nodal coordinates).
The mirror has a nominal focal length f, and a nominal focus at mirror coordinates (­f, 0, 0).
We define Focus Coordinates (FC­1.0) as a cartesian system identical to MNC coordinates but with
its origin at the focus. Thus
P (MNC) = (-f, 0, 0) + P (FC) (20)
The nominal connection between FC and STF coordinates is
P (FC) = P (STF ) (21)
However, we support the more general alignment
P (FC) = O STF (FC) +R(STF,FC)P (STF ). (22)
This equation allows us to handle the inversion of the HRMA with respect to the instrument
compartment that was originally planned for XRCF.
Associated with the MNC system are two spherical coordinate systems. Mirror Spherical Co­
ordinates measure the o# axis angle and azimuth of an incoming ray, and their pole is the MNC
+XN axis. Focal Surface Coordinates measure the o# axis angle and azimuth of an emerging ray,
with pole at the -XN axis (toward the focus). Each of these spherical coordinate systems has
an associated family of pixel plane coordinate systems (parameterized by the selected pixel size),
tangent to the pole of the sphere.
The generic Focal Plane Pixel coordinates are
FPX = FPX0 + t x # -1 s (Y N /|XN |) (23)
FPY = FPY 0 + t y # -1 s (ZN /|XN |)
where t x , t y are sign parameters which control the orientation of the image. # s is the pixel size in
radians. The actual physical pixel size at the focus, for focal length f, is
# p = f# s (24)
14

Tangent
Plane
Focal
Plane
Chip detector planes
Figure 6: The Tangent Plane Coordinate system describes the incoming rays, while the Focal Plane
Coordinates system describes the angular position of rays emerging from the mirrors. The rays
then intersect one of several inclined detector planes, causing events whose locations are described
in tiled detector coordinates.
A focal plane pixel coordinate system is defined in terms of this physical pixel size. The nominal
focal length of the mirror is then needed to convert to actual angular size. (In practice, the angular
size may be directly encoded in a WCS CDELT header parameter).
The Tangent Plane pixel coordinates are defined in the same way, but form the pixel plane for
the Mirror Spherical Coordinates. Specifically, we define the standard ASC­FP­FSC­1.0 variant
of focal plane coordinates as (t x = 1, t y = -1)
FPX = FPX0 +# -1 s (Y N /|XN |) (25)
FPY = FPY 0 -# -1 s (ZN /|XN |)
Note that in the focal plane XN is negative but we take the absolute value in the formula. The
sign on FPY is then chosen to take out the mirror inversion of the image caused by the optics. The
sign on FPX reflects the fact that we want celestial longitude to increase from right to left when
the roll angle is zero.
To repeat this in another way: (assuming zero roll angle and the default chip orientation)
. Source moves to higher RA
. Source moves to left in sky image, so sky X decreases
. FPX decreases
. Incoming photon moves to smaller MNC Y and MNC Y/|X|
. Image moves to larger MNC Y, FC Y, LSI Y, CHIP X
and
15

. Source moves to higher Dec
. Source moves up in sky image, so sky Y increases
. FPY increases
. Incoming photon moves to larger MNC Z
. Image moves to smaller MNC Z, FC Z, LSI Z, CHIP Y
We define the standard ASC­TP­MSC­1.0 version of tangent plane coordinates as
TPX = FPX0 +# -1 s (Y N /|XN |) (26)
TPY = FPY 0 -# -1 s (ZN /|XN |)
(identical to the focal plane coordinates) where here the (XN , YN , ZN ) are the unit vector in MNC
coordinates of the incoming ray. Tangent plane coordinates are defined to apply to positions on the
`sky' side of the mirror, and do not include mirror optical distortions, while focal plane coordinates
do. In practice we assume that tangent plane and focal plane coordinates are identical, and mirror
distortions are handled in the spatial variation of the centroid of the PSF.
Thus in our simple mirror model where an incoming photon coming from unit vector (X, Y, Z)
with ray unit vector (-X,-Y,-Z) is imaged to a ray with the same unit vector (-X,-Y,-Z),
the tangent plane coordinates of the incoming photon TPX, TPY are equal to the focal plane
coordinates of the imaged ray. For incoming rays the sign of X is always negative, so ­X is positive.
For XRCF, we adopt a di#erent convention. Instead of looking at the sky from the
detector, we look at the detector from the source, choosing t x = t y = -1. Then FPX is
parallel to ­ MNC Y or + XRCF Y; FPY is parallel to ­ MNC Z or + XRCF Z. Users
should be aware of the di#erence between FP coordinates at XRCF and in flight.
Table 1: Default axis orientations
Flight XRCF
­RA, +Dec +XRCF Y, +XRCF Z
+FPX, +FPY +FPX, +FPY
+MNY, ­MNZ ­MNY, ­MNZ
+FCY, ­FCZ ­FCY, ­FCZ
+LSI Y,­LSI Z ­LSI Y, ­LSI Z
+CHIP X,­CHIP Y ­CHIP X, ­CHIP Y
We define Mirror Spherical Coordinates (r, # H , #H ) in terms of mirror nodal Cartesian coordi­
16

+ +
+
+
+
Tiled Detector Plane Focal Plane (with tilted chips)
Tangent Plane (with mirror effects) Sky Plane (with aspect solution)
Figure 7: The di#erent pixel plane coordinate systems. Distorting e#ects are highly exaggerated.
nates as follows:
# # #
XN
YN
ZN
# # # =
# # #
rcos# H
r sin # H cos #H
r sin # H sin #H
# # # (27)
The angle # H is the MSC o#­axis angle and #H is the MSC Azimuth. The inverse is
r = # X 2
N + Y 2
N + Z 2
N (28)
# H = cos -1 (XN /r)
#H = arg(Y N , ZN )
The Focal Surface Coordinates (r, # F , # F ) are
# # #
XN
YN
ZN
# # # =
# # #
-rcos# F
r sin # F cos # F
r sin # F sin # F
# # # (29)
The angle # F is the FSC o#­axis angle and the angle # F is the FSC azimuth. They are related
to the MSC coordinates by
# F = # - # H , # F = #H (30)
17

HRMA
X
Y
RA
Dec
X L
Y L
Z L
SKY
DETECTOR
Dec
RA
Figure 8: Imaging the sky in LSI coordinates
The inverse is
r = # X 2
N + Y 2
N + Z 2
N (31)
# H = cos -1 (-XN /r)
#H = arg(Y N , ZN )
18

2.2.5 Pixel planes for intermediate systems
We may want to make pixel images for data in coordinate systems such as LSI and STF frames. To
do this we use the FP system and set the LSI or STF origin at the nominal focus by specifying a
physical pixel size. ASC­FP­STF­1.0 coordinates are like ASC­FP­FSC­1.0 coordinates but ignore
the possibility of defocus or misalignment (or dither) between the instrument compartment and the
mirrors. ASC­FP­LSI­1.0 coordinates further ignore the possibility that the the instrument is not
at its standard position in the focus (no SI instrument table correction) while ASC­FP­STT­1.0
coordinates artificially place the STT origin at the focus. For each of these systems,
FPX = FPX0 +# -1 sp Y
FPY = FPY 0 -# -1 sp Z (32)
where # sp is the physical pixel size and Y,Z are the LSI, STT, etc., coordinates, while the X­
coordinate is ignored.
The ASC­FP­STF­1.0 pixel coordinate system is also called 'dithered focal plane coordinates'
since it can be used to debug XRCF dither mode.
2.3 Data Analysis 2: Tangent Plane to Sky
2.3.1 Sky Pixel Coordinates
The Sky Pixel coordinate system is a translation and rotation of the Tangent Plane pixel coordinate
system to align the image with a nominal pointing direction and spacecraft roll angle. For small
aspect corrections, sky pixel coordinates (ASC­SKY­1.0) are
X = FPX0 + (TPX - FPX0) cos # - (TPY - FPY 0) sin # + AX
Y = FPY 0 + (TPX - FPX0) sin # + (TPY - FPY 0) cos # + A Y
(33)
The quantities AX and A Y are the sky frame aspect o#sets in pixels, determining the sky pixel
coordinates of the optical axis. (Note that the aspect o#sets for Einstein and Rosat were stored as
detector frame o#sets, which required applying the roll angle to; it's not clear to me why this choice
was made.)
In general when combining data over a wide range of pointing directions, (mosaicing images) we
must reproject to the nominal sky tangent plane.
2.3.2 Physical Tangent Plane coordinates
If the Tangent Plane pixel coordinates represent a position on the tangent plane to the unit sphere
at the optical axis, the Physical Tangent Plane coordinate system
# # #
X PTP
Y PTP
Z PTP
# # # =
# # #
-# s (TPX - TPX0)
# s (TPY - TPY 0)
1
# # # . (34)
19

represents the 3D vector from the center of the unit sphere to that position on the tangent plane,
which the convention that the X PTP and Y PTP axes run in the direction of increasing RA and Dec
respectively in the on­orbit case with zero roll. The PTP system is closely related to HRMA Nodal
coordinates. If the direction of the incoming ray is (XN , YN , ZN ) then
# # #
X PTP
Y PTP
Z PTP
# # # =
# # #
YN /XN
ZN /XN
1
# # # (35)
2.3.3 Physical Sky Plane coordinates
The Physical Sky Plane coordinate system for zero aspect o#set and finite roll angle is
# # #
X PSP
Y PSP
Z PSP
# # # =
# # #
X PTP cos # + Y PTP sin #
X PTP sin # - Y PTP cos #
Z PTP
# # # (36)
where # is the spacecraft roll angle.
The PTP and PSP systems are important as you need them to calculate the RA and Dec.
However they are only used in internal calculations.
2.3.4 J2000 Celestial Coordinates
One can go from Tangent Plane Physical coordinates to J2000 celestial coordinates using the in­
stantaneous pointing direction (#A , # A ). and roll angle #A .
S(#, #) = Rot(#/2 + #A , #/2 - # A , # - #A )
# # #
X PTP
Y PTP
Z PTP
# # # (37)
Use the nominal pointing direction (# 0 , # 0 ) and set the roll angle to zero if using Sky Plane Physical
Coordinates:
S(#, #) = Rot(#/2, #/2 - # 0 , # - # 0 )
# # #
X PSP
Y PSP
Z PSP
# # # (38)
From TP pixel coordinates, recall that
# # #
X PTP
Y PTP
Z PTP
# # # =
# # #
-# s (TPX - TPX0)
# s (TPY - TPY 0)
1
# # # (39)
20

CHIP
FP­LSI
FP­STT
FP­STF
CPC
LSI
STT
STF
Pixel ­­­ Physical ­­ Angular
Orient chips in 3D space
Place detector on instrument table
Move instrument table
SKY PSP CEL
Apply aspect solution
MSC
PTP
TP
Account for distorting effects of mirrors
FSC
MNC
FP
FC
Orient instrument compartment relative to
mirrors
FP
Include telescope focal length
Figure 9: Coordinate systems used in data analysis, 1: Imaging data analysis.
2.4 Simulation: Sky to Telemetry
2.4.1 Mirror Spherical Coordinates to Focal Surface Coordinates
Incoming photons with given mirror spherical coordinates (o# axis angle and azimuth) can be
described by their tangent plane pixel coordinates. To first order, these are the same as the focal
plane pixel coordinates; determining the higher order corrections is the job of ray trace simulators
such as SAOSAC. When simulating, we often will not bother with these tangent and pixel plane
coordinates but instead will work directly with mirror nodal coordinates.
2.4.2 Focal Surface or Mirror Nodal Coordinates to CPC coordinates
We then have the problem: how to transform from mirror nodal coordinates of a ray emerging
from the mirror to the detected photon position. This is harder than working in the data analysis
direction. We choose a Cartesian coordinate system (usually LSI coordinates) and calculate the
intersection of the ray line with each detector plane. There should be only one detector plane which
has an intersection within its finite bounds (i.e., the photon will only hit one of the chips). For
21

FC
Pixel Physical Angular Energy
GDC
GZO
GDP GAC l
E ;
Figure 10: Coordinate systems used in data analysis, 2: Grating data analysis.
detectors which do not include any tilted or out­of­plane chips (pre­AXAF missions), the calculation
is much easier as you can just find the LSI coordinates of the ray when it hits the focal plane.
In our more complicated situtation, as before a general point on one of the detector planes is
r = p 0 +XCPC eX + YCPC e Y (40)
where p 0 is the origin of CPC coordinates, and eX and e Y are the unit vectors along the CPC axes,
with e Z as the unit normal to the plane.
The general ray is
r = l 0 + #l (41)
where l 0 is an arbitrary point on the ray (the output of ray trace or else for approximate calculations,
the mirror node), l is the ray direction, and # labels positions along the ray. The intersection with
the plane is then at
r = l 0 +
(p 0 - l 0 ).e Z
l.e Z
l (42)
and we find CPC X and Y by taking the dot product with the CPC unit vectors,
XCPC = r.e X , YCPC = r.e Y (43)
22

2.4.3 Chip coordinates
We can then recover chip coordinates and telemetry coordinates using the equations from previous
sections.
CHIPX = XCPC /# p + 0.5 (44)
CHIPY = YCPC /# p + 0.5
2.5 Summary of coordinate systems
Each of the physical coordinate systems, representing either a 3­D position or a 2­D angular position,
has a corresponding 2­D pixel coordinate system used for display purposes and for practical storage
in cases where the third dimension is redundant.
In the following table, I give each physical system with the corresponding pixel plane systems.
`Label' is the generic label used to identify the system; data associated with this label is deemed
to satisfy the equations in this document, and revisions to the equations will imply version number
changes in the label. To isolate a specific system, extra parameters such as the chip ID are required.
The table also notes whether information is lost going from the physical to the pixel system.
Physical Name Pixel Params Info Description
System System Loss
CPC Chip Physical ASC­CHIP­1.0 Inst. ID, No Single chip
Chip ID
CPC Chip Physical ASC­TDET­1.0 Inst. ID, No Detector coords
TDET params
LSI Local SI ASC­FP­LSI­1.0 Inst. ID Yes Focal plane, o#set uncorrected
STT SI Translation Table ASC­FP­STT­1.0 Pixel size Yes Focal plane, o#set uncorrected
STF SI Translation Frame ASC­FP­STF­1.0 Pixel size Yes Focal plane, alignment uncorrected
FC Focus Coordinates ASC­FP­FSC­1.0 Pixel size Yes Focal plane
MNC Mirror Nodal ASC­FP­FSC­1.0 Pixel size Yes Focal plane
FSC Focal Surface ASC­FP­FSC­1.0 Pixel size No Focal plane
MNC Mirror Nodal ASC­TP­MSC­1.0 Pixel size Yes Tangent plane
MSC Mirror Spherical ASC­TP­MSC­1.0 Pixel size No Tangent plane
CEL Celestial ASC­SKY­1.0 Pixel size No Aspect applied
PTP Physical Tangent Plane ASC­TP­MSC­1.0 Pixel size No Tangent Plane 3D
PSP Physical Sky Plane ASC­SKY­1.0 Pixel size No Sky Plane 3D
The TDET parameters for tiled detector coordinates are arbitrary, so there can be several
parallel TDET systems for one instrument and individual TDET labels assigned for each system
must be used.
23

The pixel size used in the FP and TP systems is often equal to the detector pixel size in the
nominal focal plane, but sometimes is picked arbitrarily.
2.6 Data Analysis 3: Full treatment with misalignments
The discussion of transforming STF to MNC coordinates above is incomplete. For the full treatment,
we consider misalignment of the telescope and optical bench, and include a background spacecraft
(or lab) coordinate frame. For instance, at Chandra calibration the background coordinate frame
is the XRCF system.
We define the following new frames:
. SC, the spacecraft frame (the XRCF frame plays this role in calibration)
. DFC, the frame along whose axes the STF frame (the science instrument module) is mechan­
ically displaced and rotated.
. MFC, the mirror fiducial coordinate system along whose axes the mirror is commanded to
move.
Then a point in STF coordinates can be converted to spacecraft coords by
P (DFC) = O STF (DFC) +R(DFC,STF )P (STF )
and
P (SC) = ODFC (SC) +R(SC,DFC)P (DFC)
Here ODFC (SC) is the position in spacecraft coords of the home position of the SIM assem­
bly O STF (DFC) is the displacement of the SIM assembly relative to this home position, and
R(DFC,STF) is the rotation of the assembly relative to its home orientation. R(SC,DFC) gives
the misalignment of the spacecraft and DFC frames. We further allow an independent motion of
the mirror relative to the spacecraft frame.
P (SC) = OMFC (SC) +R(SC,MFC)P (MFC)
and
P (MFC) = OMNC (MFC) +R(MFC,MNC)P (MNC)
Here OMFC (SC) is the position of the center of rotation of the telescope in spacecraft coordinates,
R(SC,MFC) give the misalignment of the zero of pitch and yaw relative to the SC system, and
R(MFC,MNC) gives the orientation of the mirrors. OMNC (MFC) allows for a translation of the
mirror node relative to the center of rotation.
For the specific case of Chandra we have:
Parameter Name Flight XRCF
24

OMNC (MFC) Nodal o#set of center of rotn. Zero Zero?
OMFC (SC) SC coords of center of rotn. Node Node
ODFC (SC) SC coords of fiducial point OTA focus FOA
O STF (DFC) SIM frame displacement Best focus, boresight FAM displacement
R(MFC,MNC) Mirror pitch and yaw Zero HRMA pitch, yaw
R(SC,MFC) Mirror misalignment Zero? Mirror alignment
R(DFC,STF) SIM frame orientation Zero FAM rotation
R(SC,DFC) SIM frame misaligment Boresight FAM misalignment
The FITS keywords STG X,Y,Z give the SIM frame displacement, as distinct from SIM X,Y,Z
which give the SIM table position relative to the SIM frame.
3 Data Analysis Coordinate Systems ­ Gratings
We now consider objective transmission gratings placed between the mirrors and the detectors.
3.1 Data Analysis: Grating data
When we observe with the gratings, we get a dispersed spectrum with orders +1, ­1, +2, ­2, ...
and a zero­order undispersed image. The undispersed (zero­order) photons do not interact with the
gratings and we can deal with them using the same analysis as for imaging detectors. To analyse a
dispersed photon, however, we must know the location of the zero­order image as well as that of the
dispersed photon. For instance, spacecraft roll aspect must be applied to the zero­order position,
not the dispersed position.
The location of the zero order photon must be calculated relative to the Grating Node rather
than the Mirror Node. The Grating Node is on the optical axis at a distance R from the focus,
where R is the diameter of the Rowland Circle.
Each grating is defined by a grating node position and a grating pole vector which defines the
cross­dispersion direction.
3.1.1 Grating Zero Order Coordinates (GZO­1.0)
Now we pick a source, with zero order position ZO. and let the vector from the grating node to
the source zero order be S. Then define
eXZO = -S/|S|
e YZO = d 0 # eXZO /|d 0 # eXZO |
e ZZO = eXZO # e YZO
(45)
where the Grating Pole (cross­dispersion unit vector) d 0 is
d 0 = (0, - sin #G , cos #G ) (46)
25

in MNC coordinates, where #G is the angle between the dispersion direction and the spacecraft Y
axis.
This defines a cartesian orthonormal set, Grating Zero Order Coordinates, whose origin we
choose to be at G0. Di#racted photons travel in the X ZO , Y ZO plane, and the intersection of this
plane with the detector surface defines the dispersion direction.
d 0
ZO
Y
X
Z
Z ZO
X ZO
Y ZO
Y
ZO
ZO
G
F
Z
Figure 11: Grating Zero Order coordinates
The key step is calculating the GZO to FC transformation matrix. The columns of this matrix
are simply the vectors eXZ0 etc.
The GZO coordinates of the photon are then
P (GZO) = R(FC,GZO)(P (FC) -OGZO (FC)) (47)
where OGZO (LSI) are the FC coordinates of the grating node.
For an approximate treatment ( which can be in error by several pixels), we can write the zero
order coords as
ZO(FC) = (X, Y, Z) = (gX o , gY o , gZ o ) (48)
where g is the distance from G0 to the focus. Then to first order
|S| = g(1 -X o ) (49)
and
eXZO = (1, -Y o , -Z o )
e YZO = ((Z o sin #G + Y o cos #G ), cos #G , sin #G )
e ZZO = (-Y o sin #G + Z o cos #G , - sin #G , cos #G )
(50)
so that a point with FC coords (x, y, z ) will have GZO coords
P (GZO) = (x - g, (y - Y ) cos #G + (z - Z) sin #G , (z - Z) cos #G - (y - Y ) sin #G ) (51)
26

3.1.2 Grating Angular Coordinates (GAC­1.0)
Grating Angular Coordinates (GAC) are the most important system for grating analysis. The
GAC system (# r , # d ) is a 2D angular system giving longitude and latitude with respect to GZO
coordinates. The longitude coordinate, # r , is the dispersion angle and the latitude coordinate,
# d , is the cross­dispersion angle. They are defined as
# r = tan -1 # -YZO XZO # # -Y ZO /X ZO
# d = tan -1 # +ZZO
# X 2
ZO +Y 2
ZO
# # +Z ZO /X ZO
(52)
3.1.3 Grating Di#raction Coordinates (GDC­1.0)
The Grating Di#raction Coordinate system (r TG , d TG ) gives the distance in mm along the dispersion
direction and in the cross­dispersion direction. This is just related to the GAC coordinates by a
simple scaling
r TG = XR # r
d TG = XR tan # d
(53)
Here XR is the length from the grating node to the focus, which is approximately equal to the
length |S|.
G
ZO
P
d
r
Figure 12: Grating Di#raction coordinates
3.1.4 Grating Di#raction Plane Pixel Coordinates (GDP­1.1)
The Grating Di#raction Plane Pixel Coordinates GDX, GDY are defined by
GDX = GDX0-# -1 gs (Y ZO /X ZO )
GDY = GDY 0 +# -1 gs (Z ZO /X ZO ) (54)
analogously to the Focal Plane Pixel Coordinates.
They are related to the physical Grating Di#raction Coordinates by
GDX = GDX0+# -1 gs tan(r TG/XR )
GDY = GDY 0 +# -1 gs (d TG/XR ) cos(r TG/XR ) (55)
27

3.1.5 Dispersion relation
The wavelength of the di#racted photon is
# = P sin # R/m (56)
where P is the average grating period and m is the di#raction order. So
# # (P/m)(GDX -GDX0)# gs (57)
Part II
Part 2: AXAF systems
4 ACIS
4.1 Instrumental details
The ACIS instrument has 10 CCD chips. In the event list data, each is identified by an integer
from 0 to 9. Four of the chips, the imaging set, are arranged in a rough square (but individually
tilted). Six are the spectroscopic array, arranged in a line.
Chip Name CHIP ID
ACIS­I0 0
ACIS­I1 1
ACIS­I2 2
ACIS­I3 3
ACIS­S0 4
ACIS­S1 5
ACIS­S2 6
ACIS­S3 7
ACIS­S4 8
ACIS­S5 9
The CHIP ID is also called CCD ID for consistency with ASCA.
4.1.1 ACIS readout coordinates
The ACIS readout coordinate system was explained in a 1995 draft memo from J Woo. This memo
defines two coordinate systems, the ``pixel coordinate system of the readout file array, f(x,y)'', which
28

I will call the ACIS Readout Coordinates (XREAD,Y READ) with identifier AXAF­ACIS­3.0,
and the ``pixel coordinate system of the active detector image array p(x,y)'', which I will call ACIS
Chip Coordinates, (XCHIP , Y CHIP ) with identifier AXAF­ACIS­1.0 (these are the ones that
run from 1 to 1024).
ACIS Readout Coordinates may be seen in subassembly cal (SAC) data, but in flight the Chip
coordinates are calculated on board and telemetered directly. We don't normally deal with the
readout coordinates.
The two systems are related by
Y CHIP = # Y READ 1 # Y READ # 1026
Overclock 1027 # Y READ # 1030 (58)
XCHIP =
# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #
Underclock 1 # XREAD # 4
XREAD - 4 5 # XREAD # 260 (Node A)
Overclock 261 # XREAD # 337
Undefined Parallel Transfer 338 # XREAD # 340
Underclock 341 # XREAD # 344
857 -XREAD 345 # XREAD # 600 (Node B
Overclock 601 # XREAD # 677
Undefined Parallel Transfer 678 # XREAD # 680
Underclock 681 # XREAD # 684
XREAD - 172 685 # XREAD # 940 (Node C)
Overclock 941 # XREAD # 1017
Undefined Parallel Transfer 1018 # XREAD # 1020
Underclock 1021 # XREAD # 1024
2049 -XREAD 1025 # XREAD # 1280 (Node D)
Overclock 1281 # XREAD # 1357
Undefined Parallel Transfer 1358 # XREAD # 1360
(59)
The inverse transformation is
Y READ = Y CHIP
XREAD =
# # # # # # # # #
XCHIP + 4 1 # XCHIP # 256 (A)
857 -XCHIP 257 # XCHIP # 512 (B)
XCHIP + 172 513 # XCHIP # 768 (C)
2049 -XCHIP 769 # XCHIP # 1024 (D)
(60)
Unresolved questions: Is the above correct, or even useful? Do the telemetry values start at
1 or 0? Under what circumstances do we get YCHIP values of 1025 and 1026? Do those values
correspond to true active area?
29

A B C D
LL
YCHIP
XCHIP
XREAD
YREAD
Figure 13: ACIS readout nodes
4.1.2 ACIS Fast Window mode
In window mode, we get an initial set of Window Chip Coordinates (AXAF­ACIS­4.0) WX, WY
which run from 1 to 1024 and 1 to WSIZE. WSIZE is a configurable number.
There is no way to uniquely know from the telemetry what the true ACIS chip coordinates of the
photon were ­ you get an uncertainty module WSIZE. However, to calculate best guess ACIS chip
coordinates (AXAF­ACIS­1.0) we can assume that the photon is associated with a known incident
source location, resolving the uncertainties.
For non grating data, calculate the predicted AXAF­ACIS­1.0 coordinates for the incident ra­
diation, say (CX0, CY0). At XRCF, this will involve the forward calculation including the STF
coordinates of the FAM. Then set
CY1 = (CY0 / WSIZE ) * WSIZE
using integer arithmetic. Then
CHIPX = WX (61)
and
CHIPY = WY + CY 1 (62)
4.2 2­D detector coordinates: TDET parameters
The ASC­TDET­1.0 tiled detector coordinates are implemented for AXAF with the following spe­
cific systems, each of which specifies a TDET system in terms of the CHIP coordinates.
30

Table 5: Tiled Detector Plane systems
System Size X Center, Y Center Use
AXAF­ACIS­2.2 8192 x 8192 (4096.5, 4096.5) Standard
AXAF­ACIS­2.3A 32768x32768 (16384.5, 16384.5) Obsolete
Table 6: Parameters of Tiled Detector Coordinate defi­
nitions
Tiled System Chip # i # i X0 i Y 0 i H i
AXAF­ACIS­2.2 ACIS­I0 90 1 3061.0 5131.0 1
AXAF­ACIS­2.2 ACIS­I1 270 1 5131.0 4107.0 1
AXAF­ACIS­2.2 ACIS­I2 90 1 3061.0 4085.0 1
AXAF­ACIS­2.2 ACIS­I3 270 1 5131.0 3061.0 1
AXAF­ACIS­2.2 ACIS­S0 0 1 791.0 1702.0 1
AXAF­ACIS­2.2 ACIS­S1 0 1 1833.0 1702.0 1
AXAF­ACIS­2.2 ACIS­S2 0 1 2875.0 1702.0 1
AXAF­ACIS­2.2 ACIS­S3 0 1 3917.0 1702.0 1
AXAF­ACIS­2.2 ACIS­S4 0 1 4959.0 1702.0 1
AXAF­ACIS­2.2 ACIS­S5 0 1 6001.0 1702.0 1
AXAF­ACIS­2.3A ACIS­I0 90 5 12150.0 25655.0 1
AXAF­ACIS­2.3A ACIS­I1 270 5 22500.0 20535.0 1
AXAF­ACIS­2.3A ACIS­I2 90 5 12150.0 20425.0 1
AXAF­ACIS­2.3A ACIS­I3 270 5 22500.0 15305.0 1
AXAF­ACIS­2.3A ACIS­S0 0 5 800.0 8510.0 1
AXAF­ACIS­2.3A ACIS­S1 0 5 6010.0 8510.0 1
AXAF­ACIS­2.3A ACIS­S2 0 5 11220.0 8510.0 1
AXAF­ACIS­2.3A ACIS­S3 0 5 16430.0 8510.0 1
AXAF­ACIS­2.3A ACIS­S4 0 5 21640.0 8510.0 1
AXAF­ACIS­2.3A ACIS­S5 0 5 26850.0 8510.0 1
4.3 3­D chip locations: CPC to LSI transformation parameters
In the following tables we list the CPC and LSI coordinates of each corner of each chip. We give
coordinates for each of the ACIS chips in both the ACIS­I and ACIS­S LSI systems, since we may
take data from ACIS­S chips while ACIS­I is in the focus or vice versa. The systems are simply
31

o#set by 46.88mm in the Z LSI direction. The ACIS data is from ACIS­SOP­01, and the HRC data
is deduced from information provided by M. Juda.
32

Table 7: ACIS Chip corner locations in ACIS­I LSI coordi­
nates
Chip Corner CPC coords ACIS­I LSI coords
I0 LL (0.0, 0.0, 0.0) (2.361, ­26.484, 23.088)
LR (24.58, 0.0, 0.0) (1.130, ­26.546, ­1.458)
UR (24.58, 24.58, 0.0) (­0.100, ­2.001, ­1.458)
UL (0.0, 24.58, 0.0) (1.130, ­1.939, 23.088)
I1 LL (0.0, 0.0, 0.0) (1.130, 23.086, ­1.458)
LR (24.58, 0.0, 0.0) (2.360, 23.024, 23.088)
UR (24.58, 24.58, 0.0) (1.130, ­1.521, 23.088)
UL (0.0, 24.58, 0.0) (­0.100, ­1.459, ­1.458)
I2 LL (0.0, 0.0, 0.0) (1.130, ­26.546, ­1.997)
LR (24.58, 0.0, 0.0) (2.361, ­26.484, ­26.543)
UR (24.58, 24.58, 0.0) (1.130, ­1.939, ­26.543)
UL (0.0, 24.58, 0.0) (­0.100, ­2.001, ­1.997)
I3 LL (0.0, 0.0, 0.0) (2.361, 23.024, ­26.543)
LR (24.58, 0.0, 0.0) (1.131, 23.086, ­1.997)
UR (24.58, 24.58, 0.0) (­0.100, ­1.459, ­1.997)
UL (0.0, 24.58, 0.0) (1.130, ­1.521, ­26.543)
S0 LL (0.0, 0.0, 0.0) (0.744, ­81.051, ­59.170)
LR (24.58, 0.0, 0.0) (0.353, ­56.478, ­59.170)
UR (24.58, 24.58, 0.0) (0.353, ­56.478, ­34.590)
UL (0.0, 24.58, 0.0) (0.744, ­81.051, ­34.590)
S1 LL (0.0, 0.0, 0.0) (0.348, ­56.047, ­59.170)
LR (24.58, 0.0, 0.0) (0.099, ­31.473, ­59.170)
UR (24.58, 24.58, 0.0) (0.099, ­31.473, ­34.590)
UL (0.0, 24.58, 0.0) (0.348, ­56.047, ­34.590)
S2 LL (0.0, 0.0, 0.0) (0.096, ­31.042, ­59.170)
LR (24.58, 0.0, 0.0) (­0.011, ­6.466, ­59.170)
UR (24.58, 24.58, 0.0) (­0.011, ­6.466, ­34.590)
UL (0.0, 24.58, 0.0) (0.096, ­31.042, ­34.590)
S3 LL (0.0, 0.0, 0.0) (­0.011, ­6.035, ­59.170)
LR (24.58, 0.0, 0.0) (0.024, 18.541, ­59.170)
UR (24.58, 24.58, 0.0) (0.024, 18.541, ­34.590)
33

UL (0.0, 24.58, 0.0) (­0.011, ­6.035, ­34.590)
S4 LL (0.0, 0.0, 0.0) (0.026, 18.972, ­59.170)
LR (24.58, 0.0, 0.0) (0.208, 43.547, ­59.170)
UR (24.58, 24.58, 0.0) (0.208, 43.547, ­34.590)
UL (0.0, 24.58, 0.0) (0.026, 18.972, ­34.590)
S5 LL (0.0, 0.0, 0.0) (0.208, 43.978, ­59.170)
LR (24.58, 0.0, 0.0) (0.528, 68.552, ­59.170)
UR (24.58, 24.58, 0.0) (0.528, 68.552, ­34.590)
UL (0.0, 24.58, 0.0) (0.208, 43.978, ­34.590)
5 HRC
5.1 Overview
The HRC instrument has two detectors, the HRC­S and the HRC­I. The HRC­I has a single `chip' or
segment/microchannel plate pair (CHIP ID = 0, name HRC­I) while the HRC­S has three (CHIP ID =
1,2,3; name HRC­S1,S2,S3; HRC team segment designation +1, 0, ­1). We define an HRC­I chip plane
which is 16384 pixels square; (not all of these pixels correspond to actual readable values); the HRC­S
chips are 4096 x 16456 pixels The pixel size is 0.0064294 mm.
Name ID CHIP size (pix) Pixel size (µ) CHIP size (mm)
HRC­I 0 16384 x 16384 6.429 x 6.429 105.333 x 105.333
HRC­S1 1 4096 x 16456 6.250 x 6.429 25.600 x 105.802
HRC­S2 2 4096 x 16456 6.250 x 6.429 25.600 x 105.802
HRC­S3 3 4096 x 16456 6.250 x 6.429 25.600 x 105.802
5.1.1 3­D chip locations: CPC to LSI parameters
The corners of these logical chip planes are located in the LSI coordinate system as follows; these are
calculated by placing the HRC­S origin (default aimpoint) at 4.0mm to +LSI Y of the overall detector
center and using the chip sizes calculated above together with the designed S1 and S3 chip tilts in the X,Y
plane. Note that S2 is no longer centered around the detector center, since its edges are determined by the
measured gap locations.
Table 9: HRC chip (i.e. grid) corner locations in LSI coordi­
nates
Chip Corner CPC coords HRC­I,S LSI coords
HRC­I LL ( 0.000 , 0.000 , 0.000 ) ( 0.000 , 0.000 , 74.482 )
HRC­I LR ( 105.339 , 0.000 , 0.000 ) ( 0.000 , 74.482 , 0.000 )
34

HRC­I UR ( 105.339 , 105.339 , 0.000 ) ( 0.000 , 0.000 , ­74.482 )
HRC­I UL ( 0.000 , 105.339 , 0.000 ) ( 0.000 , ­74.482 , 0.000 )
HRC­S1 LL ( 0.000 , 0.000 , 0.000 ) ( 2.644 , 161.949 , ­13.167 )
HRC­S1 LR ( 26.334 , 0.000 , 0.000 ) ( 2.644 , 161.949 , 13.167 )
HRC­S1 UR ( 26.334 , 105.802 , 0.000 ) ( 0.000 , 56.180 , 13.167 )
HRC­S1 UL ( 0.000 , 105.802 , 0.000 ) ( 0.000 , 56.180 , ­13.167 )
HRC­S2 LL ( 0.000 , 0.000 , 0.000 ) ( 0.000 , 56.180 , ­13.167 )
HRC­S2 LR ( 26.334 , 0.000 , 0.000 ) ( 0.000 , 56.180 , 13.167 )
HRC­S2 UR ( 26.334 , 105.802 , 0.000 ) ( 0.000 , ­49.622 , 13.167 )
HRC­S2 UL ( 0.000 , 105.802 , 0.000 ) ( 0.000 , ­49.622 , ­13.167 )
HRC­S3 LL ( 0.000 , 0.000 , 0.000 ) ( 0.000 , ­49.622 , ­13.167 )
HRC­S3 LR ( 26.334 , 0.000 , 0.000 ) ( 0.000 , ­49.622 , 13.167 )
HRC­S3 UR ( 26.334 , 105.802 , 0.000 ) ( 2.253 , ­155.400 , 13.167 )
HRC­S3 UL ( 0.000 , 105.802 , 0.000 ) ( 2.253 , ­155.400 , ­13.167 )
5.1.2 2­D detector coordinates: TDET parameters
The ASC­TDET­1.0 tiled detector coordinates are implemented for AXAF with the following spe­
cific systems, each of which specifies a TDET system in terms of the CHIP coordinates.
Table 10: Tiled Detector Plane systems
System Size X Center, Y Center Use
AXAF­HRC­2.3I 16384 x 16384 (8192.5, 8192.5) Standard HRC­I
AXAF­HRC­2.6S 49368 x 4096 (24684.5, 2048.5) Revised HRC­S
AXAF­HRC­2.2I 32768 x 32768 (16384.5, 16384.5) Alternative
AXAF­HRC­2.2S 49152 x 4096 (24576.5, 2048.5) Incorrect, 2.5S replaces
AXAF­HRC­2.3S 16384 x 24976 (8192.5, 12488.5) Obsolete
AXAF­HRC­2.5S 49152 x 4096 (24576.5, 2048.5) Alternative
AXAF­HRC­2.4S 16384 x 16384 (8192.5, 8192.5) Old Standard HRC­S
Table 11: Parameters of Tiled Detector Coordinate defi­
nitions
Tiled System Chip # i # i X0 i Y 0 i H i
35

AXAF­HRC­2.3I HRC­I 90 1 0.0 0.0 ­1
AXAF­HRC­2.6S HRC­S1 270 1 49368.0 0.0 1
AXAF­HRC­2.6S HRC­S2 270 1 32912.0 0.0 1
AXAF­HRC­2.6S HRC­S3 270 1 16456.0 0.0 1
AXAF­HRC­2.4S HRC­S1 270 1 16384.0 1024.0 1
AXAF­HRC­2.4S HRC­S2 270 1 16128.0 6144.0 1
AXAF­HRC­2.4S HRC­S3 270 1 16384.0 11264.0 1
AXAF­HRC­2.2I HRC­I 315 1 16384.0 27969.2375 ­1
AXAF­HRC­2.5S HRC­S1 270 1 48896.0 0.0 1
AXAF­HRC­2.5S HRC­S2 270 1 32512.0 0.0 1
AXAF­HRC­2.5S HRC­S3 270 1 16640.0 0.0 1
AXAF­HRC­2.3S HRC­S1 270 1 16384.0 9216.0 1
AXAF­HRC­2.3S HRC­S2 270 1 16384.0 14336.0 1
AXAF­HRC­2.3S HRC­S3 270 1 16384.0 19456.0 1
5.2 Instrumental details
5.2.1 HRC physical layout and Tap Coordinates (HRC­6.0)
Each HRC sub­instrument contains a series of electrical 'taps' on each axis of the wire grid, which
define a continuous spatial system. The electrical axes are labelled u and v, and we will say there are
N u and N v taps on each axis, numbered starting at 0. In the internal HRC­S electronics the three
MCPs have individually numbered taps but these are combined before we see it in the telemetry.
The coarse tap positions are modified by a fine position which runs from ­0.5 to +0.5. Then we
can define an HRC Tap Coordinate System (AXAF­HRC­6.0) which runs from u = -0.5 to
u = N u - 0.5, and v = -0.5 to N v - 0.5.
5.2.2 Deriving linear tap coordinates from HRC telemetry
The instrument electronics records four numbers per axis for each event: the `center tap' (usually
the tap with the maximum voltage), and the voltages of that tap and the one on either side. These
numbers, which are the values which get coded into flight telemetry, we will refer to as HRC
36

telemetry coordinates (u ht , v ht ). The four integer components of u ht are
u ht =
# # # # # # # # #
u 0 Max tap, 0 to N u - 1 (`coarse position')
ADC1 Voltage of u coarse - 1
ADC2 Voltage of u coarse
ADC3 Voltage of u coarse + 1
(63)
and similarly for v ht ). From the telemetry coordinates we can calculate an intermediate quantity,
the fine position
u fine =
ADC3 - ADC1
ADC1 + ADC2 + ADC3
(64)
Note that
-0.5 # u fine # +0.5 (65)
We now split o# the sign of this fine position correction to obtain the tap side
s u = # +1 (u fine # 0)
-1 (u fine < 0)
(66)
and the fine position magnitude
#u = |u fine | (67)
From these we calculate the linear HRC Tap Coordinates
u = u 0 + s u C u1 (u 0 , s v )#u - s u C u2 (u 0 , s)#u 2 (68)
v = v 0 + s v C v1 (v 0 , s v )#v - s v C v2 (v 0 , s)#v 2
The C factors are called the degapping parameters; for HRC they have di#erent values for
each tap and tap side. Earlier detectors (Einstein and Rosat HRI) assumed C factors which were
independent of coarse position.
The simplest choice of the degapping parameters is to take
C u1 = C v1 = 1
C u2 = C v2 = 0,
(69)
giving us HRC raw tap coordinates,
u raw = (u coarse + u fine ) (70)
v raw = (v coarse + v fine )
These coordinates do not provide a continuous system over the detector, and an HRC image
plotted in raw coordinates contains `gaps'. With some choice of the degapping parameters, we obtain
37

a continuous (but not linear) system giving an image with no gaps. HRC degapped coordinates
(u dg , v dg ). Example values from Murray and Chappell (1989) of C used to give degapped coordinates
are
C u1 = C v1 = 1.049
C u2 = C v2 = 0.110,
(71)
so
u dg = (u coarse + 1.049u fine - 0.11u fine |u fine |) (72)
v dg = (v coarse + 1.049v fine - 0.11u fine |u fine |)
The coe#cients to be used for the HRC have not yet been determined.
5.2.3 HRC Chip Coordinates (HRC­1.1)
The HRC Telemetry Pixel Number System scales the taps by a pixel size # t = 256 to give
TELU = (u + 0.5) # # t (73)
TELV = (v + 0.5) # # t ,
integer pixel numbers which start with pixel 0 (for a tap value of ­0.5). This definition is corrected
from the one in editions 4 and earlier of this memo.
For compatibility with other data archives, we add one to these engineering coordinates and
then divide them up into individual chips to get HRC Chip Coordinates (AXAF­HRC­1.1)
CHIPX = (u - u 0 ) # # t + 0.5 = TELU -# t # (u 0 + 0.5) + 0.5 (74)
CHIPY = (v - v 0 ) # # t + 0.5 = TELV -# t # (v 0 + 0.5) + 0.5 (75)
(76)
Note that u 0 = -0.5 always. v 0 = ­0.5 for HRC­I; +0.5 for HRC­S1; 64.5 for HRC­S2; 126.5 for
HRC­S3. This corresponds to HRC­S tap boundaries at 64.5 and 126.5. It returns us to a system
in which the taps are numbered separately for each chip.
Table 12: HRC­S1 TELV, v and CHIPY
TELV Coarse v, Fine v CHIPY
0 0, ­0.5 ­255
16640 64, +0.5 16384
In this system, 1 pixel = 0.0064294 mm = 0.13 arcsec. The size of one tap is 1.646mm. Now this
coordinate system actually covers a larger area than the true possible coordinates. For instance,
38

v taps 0 and 1 for HRC­S1 are missing, so the lowest possible v coordinate in the telemetry for
HRC­S is 1.5 (corresponding to tap 2 with fine position ­0.5) but even this does not correspond to
a valid detected event position. Nevertheless, we will define our logical coordinate system to cover
the range of v coordinates starting at CHIPX, CHIPY = 0.5 (lower left corner of first pixel) which
corresponds to HRC­S u, v = (­0.5, +0.5). Version 3.0 of this document defined chip coordinate
system AXAF­HRC­1.0 which did not cover this full logical range and had a slightly di#erent origin;
the current system is denoted AXAF­HRC­1.1.
The center of HRC­S2 is then at (u, v) = (7.5, 95.5) and the gaps between the MCPs are at v
values of 62.96 to 64.12 and 124.88 to 126.04. In earlier work I arbitrarily set the chip boundaries
at 63.0 and 126.0 so that each chip has a length of 63.0 taps, but on recommendation from M. Juda
I have adjusted the chip sizes so that the true chip gaps fall on the logical chip boundaries.
Table 13: HRC electronically meaningful coordinate
ranges
Chip v 0 u v CHIPX CHIPY TELU TELV
HRC­I ­0.5 0.0 to 63.0 0.0 to 63.0 0.5 to 16128.5 0.5 to 16128.5
HRC­S1 0.5 0.0 to 15.0 1.5 to 64.5 128.5 to 3968.5 256.5 to 16384.5
HRC­S2 64.5 0.0 to 15.0 64.5 to 126.5 128.5 to 3968.5 0.5 to 15872.5
HRC­S3 126.5 0.0 to 15.0 126.5 to 190.5 128.5 to 3968.5 0.5 to 16384.5
Now let's look at the boundaries on HRC­S and HRC­I more closely. We keep extra figures for
self­consistency only assuming a tap scale of 1.6460 exactly, and measure positions starting at the
physical position corresponding to chip pixel position 0.5 (bearing in mind this may be outside of
the wire grid).
39

I used the following information on the HRC­S:
Tap size is 1.646 mm (M. Juda)
16384 pixels = 105.344 mm: logical chip size = tap size 1.646 mm x 64 taps
MCP physical size 100.000 mm x 27.000 mm from 'TOP MCP COORDINATES' drawing
Total logical length is 3 x 105.344 mm
Total physical length is 3 x 100.000 mm + 2 x gap size = 1.905 mm.
S2 Center is center of both total logical and total physical length.
Coating extends 94.5mm on outer MCPs and 16mm wide (M. Juda)
Post XRCF revision 1: (M Juda Oct 97)
S1 TELU range from 600 to 3496, TELV from 1190 to 16250 with
CsI limit at 1613.
S2 TELU from 600 to 3488, TELV from 16990 to 32261
S3 TELU from 600 to 3504, TELV from 32925 to 48110 with CsI
limit at 47650.
The coating strip is at TELU = 2660 (S1), 2670 (S2), 2670 (S3)
with the 'T' strip at TELV from 22780 to 27670.
More accurate post XRCF data: MCP edges at TELV = 16428, 17012, 32244, 32932.
This information leads to the following MCP layout :
Area covered by pixel numbering
Physical size of MCP
Active area of MCP
Coated area of MCP
16384 pixels 16384 pixels 16384 pixels
4096 pixels
Figure 14: Relationship of HRC­S pixels to the physical instrument.
Table 14: HRC­S boundaries (Revised Jan 99)
40

Boundary TELV Tap v Pos, mm Seg No. YCPC , mm CHIPY, pix
S1 Logical Left Edge ­12 0.000 1 0.000 0.5
S1 TELV = 0 0 ­0.50 0.077 1 0.077 12.5
S1 Electronic L Edge 512 1.50 3.369 1 3.369 524.5
S1 MCP Left Edge 1196 4.17 7.767 1 7.767 1208.5
S1 Active Left Edge 1664 6.00 10.776 1 10.776 1676.5
S1 Active Right Edge 16404 63.58 104.545 1 105.545 16416.5
S1 MCP Right Edge 16444 63.73 105.802 1 105.802 16456.5
S1 Logical Right Edge 16444 63.73 105.802 1 105.802 16456.5
Gap left edge 16444 63.73 105.802 1 105.802 16456.5
Gap right edge 16459.9 63.80 105.802 1 105.802 16456.5
S2 Logical Left Edge 16460 63.80 105.802 2 0.000 0.5
S2 MCP Left Edge 16900 65.51 108.631 2 2.829 440.5
S2 Active Left Edge 17060 66.14 109.660 2 3.858 600.5
S2 Center 24576 95.50 157.983 2 52.181 8116.5
S2 Aimpoint 25198 97.93 161.982 2 56.180 8738.5
S2 Active Right Edge 32165 125.14 206.776 2 100.974 15705.5
S2 MCP Right Edge 32250 125.48 207.322 2 101.520 15790.5
S2 Logical Right Edge 32916.0 128.08 211.604 2 105.802 16456.5
S3 Logical Left Edge 32916.0 128.08 211.604 3 0.000 0.5
Gap left edge 32916.4 128.08 211.604 3 0.000 0.5
Gap right edge 32930.0 128.13 211.604 3 0.000 0.5
S3 MCP Left Edge 32930 128.13 211.604 3 0.000 0.5
S3 Active Left Edge 32940 128.17 211.668 3 0.064 10.5
S3 Active Right Edge 47605 185.46 305.955 3 94.351 14675.5
S3 MCP Right Edge 48117 187.46 309.247 3 97.643 15187.5
S3 Electronic R Edge 48640 189.50 312.604 3 101.000 15710.5
S3 Logical Right Edge 49386 192.41 317.406 3 105.802 16456.5
Boundary TELU Tap u XCPC , mm CHIPX, pix
MCP Edge ­1.620 ­0.700 ­286.2
TELU = 0 0 ­0.500 0.000 0.5
Logical Edge ­0.500 0.000 0.5
Active edge 560 3.500 560.5
Coating Edge 1.843 3.748 600.0
41

Center 2048 7.500 12.800 2048.5
Strip Edge 9.930 16.688 2670.0
Coating Edge 13.172 21.875 3500.0
Active edge 3536 22.10 3536.5
Logical Edge 15.500 25.600 4096.5
MCP Edge 15.937 26.300 4208.5
U
V
Z
STT
Y
STT
Figure 15: HRC­I pixel axes.
Table 15: HRC­I boundaries
Boundary Tap u or v XCPC or YCPC , mm CHIPX or CHIPY, pix
Logical Edge ­0.500 0.000 0.5
MCP Edge 1.123 2.672 416.1
Active Area 3.250 6.172 960.4
Coating Edge 4.161 7.672 1193.7
Center 31.500 52.672 8192.5
Coating Edge 58.839 97.672 15191.3
Active Area 59.750 99.172 15424.6
MCP Edge 61.877 102.672 15968.9
Logical Edge 63.500 105.344 16384.5
The CPC coordinates run from 0.0 to 26.33 (XCPC for HRC­S) and from 0.0 to 105.3 (YCPC for
HRC­S and both axes for HRC­I).
The active area of each microchannel plane is smaller, and the area coated with photocathode
42

is smaller still. For HRC­I, the chip is 100 x 100 mm, with a 93 x 93 mm active area and a 90 x
90 mm coated area. For HRC­S, each chip is 100 x 27 mm, the active area is 100 x 20 mm, and
the coated area is 94.5 x 16.0 mm. except for HRC­S2 where the coated area is 100 x 16 mm.
Using these numbers, we derive the locations of the various areas in CPC (mm) and Chip (pixel)
coordinates listed above.
43

6 The SIM
6.1 Chip orientation summary tables
The following table gives the Euler rotation angles for the chips. The LSI to CPC transformation
is more intuitive; the first angle # indicates the tilt with respect to the LSI plane (with HRC­I
having # = 180 to indicate being completely flipped over, another way of expressing the di#erent
handedness of its axes); and the third angle # indicates the rotation of the chip in the LSI Y,Z
plane relative to chip I0.
Table 16: Euler angles in degrees for CPC to LSI coor­
dinates
Chip CPC to LSI LSI to CPC
I0 Rot(180, 92.875, 177.129 ) Rot( 2.871, 92.875, 0.0 )
I1 Rot( 0, 92.872, 182.869 ) Rot( ­2.869, 92.872, 180.0)
I2 Rot(180, 87.125, 177.131 ) Rot( 2.869, 87.125, 0.0)
I3 Rot( 0, 87.128, 182.871) Rot(­2.871, 87.128, 180.0)
S0 Rot( 90.0, 90.0, 179.088) Rot( 0.912, 90.0, 90.0 )
S1 Rot( 90.0, 90.0, 179.419) Rot( 0.581, 90.0, 90.0 )
S2 Rot( 90.0, 90.0, 179.751) Rot( 0.249, 90.0, 90.0 )
S3 Rot( 90.0, 90.0, 180.082) Rot(­0.082, 90.0, 90.0 )
S4 Rot( 90.0, 90.0, 180.424) Rot(­0.424, 90.0, 90.0 )
S5 Rot( 90.0, 90.0, 180.746) Rot(­0.746, 90.0, 90.0 )
HRC­I Rot(­135.0, 90.0, 0.0) Rot( 180.0, 90.0, ­45.0)
HRC­S1 Rot(0.0, 90.0, 181.426) Rot(­1.426, 90.0, 180.0)
HRC­S2 Rot(0.0, 90.0, 180.0) Rot( 0.0, 90.0, 180.0)
HRC­S3 Rot(0.0, 90.0, 178.778) Rot( 1.222, 90.0, 180.0)
6.2 Relative positions of instruments
The location of the origins of the LSI system for each instrument are given in STT coordinates in
the table below.
Table 17: Location of instrument origin on Translation
Table
Values of P STT (S), i.e. o#sets S - #
ACIS origin (0.0, 0.0, 237.4)
HRC­I origin (0.15, 0.0, ­126.6)
HRC­S origin (0.10, 0.0, ­250.1)
44

6.3 Aimpoints
Several named default aimpoints are defined for the various AXAF instruments. The table below
gives the STF coordinates of the instrument table origin for each of these aimpoints. Most obser­
vations will use one of these default aimpoints, but in general the SIM can be moved in X and Z to
any aimpoint.
Table 18: SIM position o#sets for nominal focus positions
Values of P STF (#)
AI1 ACIS­I o#set (0.0, 0.0, ­237.4)
AI2 ACIS­I o#set (0.0, 0.0, ­233.9)
AS1 ACIS­S o#set (0.0, 0.0, ­190.5)
HI1 HRC­I o#set (­0.15, 0.0, 126.6)
HS1 HRC­S o#set (­0.10, 0.0, 250.1)
7 The HRMA (flight)
7.1 HRMA nodal coordinates
The conversion from STF (instrument compartment) to MNC (HRMA nodal) coordinates requires
knowledge of the focal length, which is 10061.0 mm.
# # #
X STF
Y STF
Z STF
# # # +
# # #
f
0
0
# # # =
# # #
XN
YN
ZN
# # # (77)
In flight, the orientation of the SIM with respect to the HRMA is fixed. In the flight nominal
configuration, one of the SIs has its nominal focal point at the telescope focus. However, the SIM
can be moved so that the nominal focal point and the telescope focus do not coincide (general flight
configuration).
7.2 Focal and Tangent plane systems
We define the following focal plane pixel systems, together with their usual purpose (although one
may still use e.g. FP­2.0 with XRCF ACIS data, etc.). The coordinate systems are defined in terms
of their physical pixel size at the nominal flight focal length f = 10061.0 mm. The corresponding
angular sizes are also given, but the actual angular size will be di#erent for XRCF/HRMA and
XRCF/TMA data.
We define systems for the nicely rounded pixel sizes and for the accurate actual pixel sizes.
We've decided to use the mean actual pixel sizes for the flight system. Thus, the size of an HRC­I
45

pixel at the nominal focal distance of 10061.0 mm is 0.132 arcseconds, so 0.132 arcseconds is defined
as the FP­2.1 pixel size (even if the detector is moved well o# focus). The HRC­S detector has a
maximum linear extent of over 48500 pixels. When making sky coordinates for such a detector, we
must define a square pixel plane (to allow for the roll angle). We choose a 65536 pixel sided plane,
defined as FP­2.3. This plane has a total of 4.3 gigapixels, slightly more than can fit in a 4­byte
signed integer. We therefore recommend not making full resolution full frame image files in this
coordinate system.
Here are the current flight pixel systems:
System # sp # s0 t x t y FPX0, FPY0 Purpose
(mm) (arcsec)
AXAF­FP­1.1 0.0240 0.492 +1 ­1 4096.5 Flight ACIS, actual pixel
AXAF­FP­2.1 0.006429 0.132 +1 ­1 16384.5 Flight HRC­I, actual pixel
AXAF­FP­2.3 0.006429 0.132 +1 ­1 32768.5 Flight HRC­S
Here are the historically used pixel systems:
System # sp # s0 t x t y FPX0, FPY0 Purpose
(mm) (arcsec)
AXAF­FP­1.0 0.0244 0.5 +1 ­1 4096.5 Flight ACIS
AXAF­FP­1.1 0.0240 0.490 +1 ­1 4096.5 Flight ACIS, actual pixel
AXAF­FP­2.0 0.0061 0.125 +1 ­1 16384.5 Flight HRC
AXAF­FP­2.1 0.0064294 0.132 +1 ­1 16384.5 Flight HRC­I, actual pixel
AXAF­FP­2.2 0.006429 0.132 +1 ­1 23000.5 Flight HRC­S, test for long int constraint
AXAF­FP­2.3 0.006429 0.132 +1 ­1 25000.5 Flight HRC­S
AXAF­FP­3.0 0.0244 0.5 +1 +1 4096.5 XRCF ACIS, obsolete
AXAF­FP­3.1 0.0244 0.5 ­1 ­1 4096.5 XRCF ACIS
AXAF­FP­3.2 0.0240 0.483 ­1 ­1 4096.5 XRCF ACIS, actual pixel size
AXAF­FP­4.0 0.0061 0.125 +1 +1 16384.5 XRCF HRC, obsolete
AXAF­FP­4.1 0.0061 0.125 ­1 ­1 16384.5 XRCF HRC
AXAF­FP­4.2 0.0064294 0.129 ­1 ­1 16384.5 XRCF HRC, actual pixel size
AXAF­FP­4.3 0.0064294 0.129 ­1 ­1 32768.5 4196.5 XRCF HRC, new
AXAF­FP­5.1 0.00240 0.490 +1 +1 2048.5 ACIS­2C
AXAF­FP­5.2 0.0024 0.483 +1 +1 2048.5 ACIS­2C
AXAF­FP­6.1 0.00643 0.132 +1 +1 4096.5 HSI
AXAF­FP­6.2 0.00643 0.129 +1 +1 4096.5 HSI alternate
(The total image size should be 2*(FPX0­0.5)).
However, software should also support generic ASC­FP/TP coordinates with arbitrary pixel size
46

and sign choices.
7.2.1 Mosaicing XRCF images
We want to make mosaiced images which show the point spread function at di#erent o# axis angles
over the whole field of view. The AXAF­FP­2.0 or ­2.1 systems are therefore of interest. However,
a single XRCF ACIS­2C image covers only a small part of the field. We don't want to make a
32768 square image when only a 4096 square part is used. As long as we stay in event list format,
there is no storage problem, but in an image we should make only the minimal subset and store
the location within the larger image as a WCS. The AXAF­FP­5.1,6.1 systems can be used for this
purpose.
7.3 Angular systems
7.3.1 HRMA Left Handed Spherical Coordinates (AXAF­HSC­1.1)
This system is used to express o#­axis angles. We define HRMA Spherical Coordinates (r, # H , #H )
in terms of HRMA nodal Cartesian coordinates as follows:
# # #
XN
YN
ZN
# # # =
# # #
+rcos#H
-r sin # H cos #H
r sin # H sin #H
# # # (78)
The inverse is
r = # X 2
N + Y 2
N + Z 2
N (79)
# H = cos -1 (XN /r)
#H = arg(-YN , ZN )
This coordinate system is also used for input to the XRCF Test Database; it is a LEFT HANDED
coordinate system. The XRCF test database immediately converts these to pitch and yaw. The
north pole of this system is the center of the forward aperture of the HRMA A0; # H measures the
o#­axis angle of the incoming ray and #H measures its azimuth in the YN , ZN plane such that for an
observer at XRCF standing by the SI and looking at the XSS, #H = 0 is to the left and #H = #/2
is vertically downwards.
Then the forward aperture of the HRMA, A0, has HRMA nodal coordinates (a, 0, 0) and HSC
coordinates (a, 0, 0). The focal point F of the HRMA has HRMA nodal coordinates (-f, 0, 0) and
HSC coordinates (f, #, 0).
The HSC­1.1 system actually makes most sense when looking in the sky plane. Consider a source
in the sky when roll angle is zero:
. The FPX, FPY and X,Y axes are aligned.
47

. RA, Dec are parallel to ­X, +Y. (since RA increases toward the left looking at the sky).
. The +X, +Y axes are parallel to ­SCY, +SCZ. A photon from larger Dec and smaller RA
than the field center comes from the +X, +Y direction. It enters the telescope from ­SCY,
+SCZ, and therefore has HSC coordinate azimuth between +0 and +90 deg. Thus, the HSC
azimuth is just the conventional plane polar coordinate angle in the +FPX, +FPY tangent
plane coordinate system.
. After passing through the (ideal) mirror origin, the photon leaves from +SCY, ­SCZ and lands
at +LSI Y, ­LSI Z. (The HSC coordinates of the outgoing photon have azimuth between 180
and 270, and o# axis angle between 90 and 270, but this is not normally of interest.)
7.3.2 HRMA Right Handed Spherical Coordinates (AXAF­HSC­1.2)
To satisfy my desire for using left handed systems as little as possible, I defined HSC­1.2 to be the
same as HSC­1.1 except that the azimuth increases in the opposite direction: #HR = 0 is to the left
and #HR = #/2 is vertically upwards. This system has not been used and is now deprecated.
# # #
XN
YN
ZN
# # # =
# # #
rcos# HR
-r sin # HR cos #HR
-r sin # HR sin #HR
# # # (80)
The inverse is
r = # X 2
N + Y 2
N + Z 2
N (81)
# HR = cos -1 (XN /r) = # H
#HR = arg(-YN , -ZN ) = -#H
7.3.3 HRMA rotation coordinates (Pitch and Yaw) (AXAF­HSC­3.0)
A third choice of pole is the +YN axis, whose latitude­like coordinates # z is called yaw, and whose
azimuthal coordinate # y is called pitch. The mapping between pitch and yaw coordinates and
HRMA coordinates is
# # #
XN
YN
ZN
# # # =
# # #
+r cos # z cos # y
+r sin # z
-r cos # z sin # y
# # # (82)
or
# z = sin -1 (Y N /r) (83)
# y = arg(XN , -ZN )
48

The rotation matrix from HSC­1.2 right­handed spherical coords is
R(HSC­1.2,HSC­3.0) = Rot(#, #/2, #). (84)
The motivation for this coordinate system is its relationship to the commanded pitch and yaw
of the HRMA at XRCF. We call the pitch and yaw coordinates of the XSS # y (XSS) = # y0 and
# z (XSS) = # z0 . To put the XSS at these coordinates, the HRMA must be yawed # z0 to the left
and its aperture pitched # y0 downward.
7.3.4 HRMA Source coordinates (AXAF­HSC­2.1)
The HRMA Source Coordinate system (r, az, el) is a pseudo RA, Dec system that gives the `sky'
position of the source as seen by the HRMA. Unlike HRMA spherical coordinates, they have a
pole at -ZN rather than XN . The relationship between Source coordinates and HRMA nodal
coordinates is:
# # #
XN
YN
ZN
# # # =
# # #
+r cos el cos az
-r cos el sin az
-r sin el
# # # (85)
Coordinates az =0, el = 0 refer to a source on the HRMA axis (center of the field of view);
positive el gives a source above the HRMA axis (top of the field of view), while positive az gives a
source to the left of the center of the field of view. (HSC­2.0 had az going the other way, creating
a left­hand system). The inverse function is
el = sin -1 (-ZN /r) (86)
az = arg(XN , -YN )
The rotation matrices from the other HRMA angular systems are
R(HSC­1.2,HSC­2.1) = Rot(#/2, #/2, #); R(HSC­3.0,HSC­2.1) = Rot(#/2, #/2, 3#/2). (87)
8 HETG and LETG
The Grating Node is on the optical axis at a distance R from the focus, where R is the diameter of
the Rowland Circle. The nominal Rowland Circle diameter is quoted as 8650.0 mm [1], Appendix
A, p. 11; 8633.69 mm [1], Drawing 301331, Sheet 3; and 8636.00 mm [3]. I will adopt the value
from the drawing, i.e. 8633.69mm. (However View F on the same drawing shows the Rowland
circle intercepting the X axis at XA = 372.116 corresponding to R = 8643.11 mm. This is an error
caused by confusing H0 and H1 when measuring the location of the OTG origin.)
The MNC coordinates of the Grating Node G0 in flight are
49

# # #
XN (G0)
YN (G0)
ZN (G0)
# # # =
# # #
-1431.81 0
0
# # # (88)
The exact value is di#erent at XRCF. We use the same physical pixel size as for the detector
systems, which correspond to somewhat di#erent angular sizes than the imaging case because we
are measuring angles from G rather than H0. However, in the case of ACIS, we hope to get extra
resolution by using the sub­pixel dither of the detector relative to the sky coordinate zero order
position. So, in the GDP­1.1 system we have chosen the GDP pixel size for ACIS and HRC to be
the same 6 micron value. (In an earlier version of this memo, the GDP­1.0 system was described
with 24 micron ACIS GDP pixel sizes. That system was never implemented.)
What is the extent of the GDP coordinate system? In the case of HRC­S, our worst case is
a zero order at an extreme end of the 3­chip array. In that case, you could be as many as 49152
pixels down the dispersion coordinate. This is such a pathological case that we have decided not to
build our definitions around it. A normal case would be a zero order in the center, with dispersion
coordinates up to plus or minus 25000 pixels or so. As a compromise to support most o# axis cases,
we adopt pixels running from 1 to 65536 in the dispersion direction, giving 32768 pixels on either
side of zero order. In the cross dispersion direction, with HRC­I as the detector, we might have
photons up to 16384 pixels from the zero order in principle, and 8192 even if the source is centered in
the detector. For ACIS, the extreme range is 32678 super­resolved pixels in the dispersion direction
and 24800 pixels in the cross dispersion direction. For both ACIS and HRC, the GDP­1.1 system is
sized to be 65536 pixels in the dispersion direction and 32678 pixels in the cross dispersion direction,
with the zero order position at (32768.5, 16384.5). However, we recommend that in normal use the
32768 x 4096 subarray extending from (16384,14336) to (49152,18432) be used. This subarray is
defined as AXAF­GDP­1.2; it has the same pixel size as GDP­1.1.
Table 21: GDC pixel image centers
Instrument System GDX0, GDY0 Image size
ACIS,HRC GDP­1.1 32768.5, 16384.5 65536 x 32768
ACIS,HRC GDP­1.2 16384.5, 2048.5 32768 x 4096
Table 22: GDP Pixel Sizes (assuming flight Rowland ra­
dius)
Instrument System Size at Focal Plane Angular Size
# p , # gp (mm) # g s (arcsec)
ACIS GDP 0.006430 0.154
50

HRC GDP 0.006430 0.154
The average grating periods for the three gratings are given in the table below.
Table 23: Grating properties
Instrument P #G (deg)
HETG 2000.0 š A 5.0
METG 4000.0 š A ­5.0
LETG 9921.0 š A +0.0
9 XRCF
9.1 XRCF coordinates
For much of XRCF, the HRMA sits aligned with the Facility Optical Axis in its 'rest' position, but
it can be tilted in two axes. We therefore need to distinguish between facility (XRCF) coordinates
and mirror (MNC) coordinates.
In the XRCF, we have the HRMA mounted on two axes ­ it can change its yaw (azimuth)
and pitch (elevation), or equivalently the polar angle and polar azimuth. The instrument (SI) is
mounted on either the FAM (Five Axis Mount) or the HXDS (HRMA X­ray Detection System).
In the default configuration C0, the HRMA axis and the SI are aligned with the FOA (Facility
Optical Axis) and the SI nominal focus S is located at the actual focus F. The X­ray Source System
(XSS) is fixed in the XRCF frame and lies on the positive XXRCF axis (see below).
The origin of XRCF coordinates is at the HRMA CAP (H1).
10 ACIS­2C
The ACIS­2C instrument is used for ground calibration at XRCF. The two chips are similar to the
flight ACIS chips, and will frequently be used in Fast Window mode.
Table 24: Tiled Detector Plane systems ­ ACIS­2C
System Size X Center, Y Center Use
AXAF­ACIS­2.4 4096 x 2048 ( 2048.5, 1024.5 ) XRCF ACIS­2C
51

Table 25: Parameters of Tiled Detector Coordinate defi­
nitions
Tiled System Chip # i # i X0 i Y 0 i H i
AXAF­ACIS­2.4 ACIS­C0 0 1 304.0 512.0 1
AXAF­ACIS­2.4 ACIS­C1 0 1 1536.0 512.0 1
The ACIS­2C chip centers are separated by approximately 63.0 mm.
52

Table 26: ACIS­2C chip corner locations in ACIS­2C LSI
coordinates
Chip Corner CPC coords ACIS­2C LSI coords
C0 LL (0.0, 0.0, 0.0) (0.000, 47.57, 11.10)
LR (24.58, 0.0, 0.0) (0.000, 22.99, 11.10)
UR (24.58, 24.58, 0.0) (0.000, 22.99, ­13.48)
UL (0.0, 24.58, 0.0) (0.000, 47.57, ­13.48)
C1 LL (0.0, 0.0, 0.0) (­0.342, ­15.18, 10.45)
LR (24.58, 0.0, 0.0) (­0.342, ­39.76, 10.45)
UR (24.58, 24.58, 0.0) (­0.342, ­39.76, ­14.13)
UL (0.0, 24.58, 0.0) (­0.342, ­15.18, ­14.13)
11 HSI
Just as ACIS­2C is like a small ACIS, HSI is like a small HRC.
HSI has 16 taps on its side and an area of 18 mm.
Table 27: HSI boundaries
Boundary Tap u or v XCPC or YCPC , mm CHIPX or CHIPY, pix
Logical Edge ­0.500 0.000 0.5
MCP Edge 2.032 4.168 648.7
Active Area ?
Coating Edge ?
Center 7.500 13.168 2048.5
Coating Edge ?
Active Area ?
MCP Edge 13.970 22.168 3448.3
Logical Edge 15.500 26.336 4096.5
Table 28: Parameters of Tiled Detector Coordinate defi­
nitions
Tiled System Chip # i # i X0 i Y 0 i H i
AXAF­HSI­2.3 HSI 0 1 0.0 0.0 1
53

Table 29: HSI chip corner locations in HSI LSI coordi­
nates
Chip Corner CPC coords HSI LSI coords
HSI LL (0.0, 0.0, 0.0) (0.000, ­13.17, ­13.17)
LR (26.34, 0.0, 0.0) (0.000, 13.17, ­13.17)
UR (26.34, 26.34, 0.0) (0.000, 13.17, 13.17)
UL (0.0, 26.34, 0.0) (0.000, ­13.17, 13.17)
11.1 Instrument origins
Table 30: Location of instrument origin on Translation
Table, XRCF
Values of P STT (S), i.e. o#sets S - #
ACIS origin (0.0, ­5.35, 67.20)
HRC­I origin (0.0, 0.0, 517.988)
HRC­S origin (0.0, 0.0, 637.480)
HSI origin (0.0, 0.0, 0.0)
ACIS­2C origin (0.0, 0.0, 65.002) (Phase F)
ACIS­2C origin (0.0, ­4.33 133.310) (Phase G)
In addition, the raw FAM values must be adjusted by (87.92, 12.94, 7.55) to get FAM coordinates
relative to the DFC origin.
11.2 HRMA motion
In the nominal configuration CN, the HRMA is tilted but the SI is moved so that its normal X LSI
axis remains coincident with the HRMA optical axis XXRCF . In the most general configuration the
HRMA and SI are both moved relative to the XRCF but the X LSI axis is not made to coincide
with the XXRCF axis.
WARNING: The terms `on­axis' and `o#­axis angle' are correctly used to refer to an angle
relative to the optical axis of the HRMA. However, at XRCF they are sometimes used to mean
an angle relative to the Facility Optical Axis, which could lead to confusion. I will always use the
concept of on and o# axis to refer to the HRMA optical axis.
54

The Y LSI , Z LSI plane contains the science instrument, which returns coordinate values of events
measured in detector pixels CHIPX, CHIPY from each of several discrete planar `chips' (or MCPs
in the case of HRC). In data analysis we consider two complementary problems:
. The forward case: Given an XRCF configuration, at which chip CHIP ID and chip pixel
CHIPX, CHIPY will the XSS photons fall? In other words, where do we expect the image to
be?
. The inverse case: Given a photon landing on chip CHIP ID and chip pixel CHIPX, CHIPY,
from which direction (o#­axis angle and azimuth) did the incoming photon approach the
HRMA?
11.2.1 XRCF to MNC transformation
We assume that the roation is around the point OMNC = (0, 0, 0) in MNC coordinates, or (XH, 0,
0) in XRCF coordinates, where XH = 9.42.
In terms of the pitch and yaw, the transformation between XRCF and MNC coordinates is given
by the translation vector (XN (H1), 0, 0) = (XH, 0, 0), and the Euler rotation matrix
R(XRCF,MNC) = Rot(# z , # + # y , #) (89)
or explicitly
XN = (XXRCF -XH) cos # z cos # y + YXRCF sin # z cos # y - ZXRCF sin # y (90)
YN = (XXRCF -XH) sin # z - YXRCF cos # z
ZN = -(XXRCF -XH) cos # z sin # y - YXRCF sin # z sin # y - ZXRCF cos # y
and the inverse
XXRCF = XH +XN cos # z cos # y + YN sin # z - ZN cos # z sin # y (91)
YXRCF = XN sin # z cos # y - YN cos # z - ZN sin # z sin # y
ZXRCF = -XN sin # y - ZN cos # y
Note that for zero pitch and yaw,
XN = (XXRCF -XH) (92)
YN = -YXRCF
ZN = -ZXRCF
reflecting the fact that the mirror is installed upside down at the XRCF.
So in particular if the focal point PN (F ) = (-f, 0, 0) then
PXRCF (F ) = (XH - f cos # z cos # y , -f sin # z cos # y , f sin # y ) (93)
55

For some reason the Ball SER approximates this as
PX (F ) # (-f cos # (# 2
z + # 2
y ), -f sin # z , f sin # y ) (94)
which is OK for small # y ,# z .
The XSS is at HRMA nodal coordinates
# # #
XN (XSS)
YN (XSS)
ZN (XSS)
# # # =
# # #
+LS cos # z cos # y
+LS sin # z
-LS cos # z sin # y )
# # # (95)
11.3 HXDS Motion
The HXDS stage moves in XRCF X, Y and Z. For a tilted mirror, one must convert from XRCF
X, Y and Z to MNC X, Y and Z to obtain the STF to MNC translation­rotation matrix.
11.4 FAM Motion
Treatment of the FAM is complicated by the fact that the FAM axes are inverted with respect to
the mirror axes.
The FAM rotates and translates the origin of STF coords, O STF , relative to the facility optical
axis, OXRCF . The three translational axes of the FAM define a (moving) FAM coordinate system
with origin at O STF . Move the FAM to the FOA so that the STF and FC origins coincide; then the
FAM coordinate system is coincident with a (fixed) DFC (Default FAM Coordinate) system which
is almost aligned with the XRCF axes. We record the transformation from DFC to XRCF axes as:
P (XRCF ) = R(DFC,XRCF )P (DFC)
where the matrix is a misalignment matrix close to unity. Let the FAM position readouts at
OXRCF be F0. Now move the FAM to an arbitrary position with FAM readouts F. Then the DFC
coordinates of the STF origin are
O STF (DFC) = F - F0
and the DFC coordinates of a position with FAM coords P(FAM) are
P (DFC) = O STF (DFC) +R(DFC,FAM)P (FAM).
Now the instrument is mounted on the LASSZ (a simulator for the SIM translation table). Let
the LASSZ reading be L, and the LASSZ reading when O STT = O STF be L0. The sign of LASSZ is
such that L increases with XRCF +Z, FAM +Z, or SC ­Z, STF ­Z. A point with LSI coordinates
P(LSI) has FAM coordinates
P (FAM) = -P (LSI) -O LSI (STT ) + L - L0
56

So from this we can go from P(LSI) to P(FAM) and hence P(XRCF).
Let's consider a FAM move with the HRMA held constant. At the start, the beam is at
chip coords CX0, CY0, and the focal plane coordinates of CX0, CY0 are the central values FX0,
FY0. Move the FAM to larger Y,Z FAM readouts. The FAM moves to larger +Y, +Z in XRCF
coordinates; the beam will then move to smaller chip coords CX, CY (the detector moved up and
to the right, so the beam, which stays put, lands nearer to the lower left of the detector). of course
the beam's FP coords remain the same. The FP coords of the old chip position CX0, CY0 have
moved to larger values of FX0, FY0 (in the default XRCF focal plane system with signs ­1,­1).
The pixlib routine pix setup mirror gets its arguments in FAM coordinates. The mirror angles
are also chosen wrt XRCF signs. Specifically, FAM angles of all zero result in an STF to DFC
transformation which inverts STF relative to DFC.
The HXDS stage axis misalignment is from a 1996 Dec 27 memo of Ian Evans.
Table 31: Assumed Misalignments
Stage Angles (arcmin)
HXDS ( 0.0, 1.6, ­16.4 )
FAM ( 0.0, 0.0, 0.0 )
11.5 FAM Motion, old copy
The FAM data file returns relative XRCF coordinates which we will designate as P FAM (XRCF ) =
X FAM , Y FAM , Z FAM and absolute XRCF orientations # x , # y , # z . We need to convert this to an STF­
MNC translation­rotation matrix. Let us assume we have converted the FAM variables to units of
mm and radians. Then from Part 1 of this memo, we need to find the parameters of
P (MNC) = O STF (MNC) +R(STF,MNC)P (STF ). (96)
These are
O STF (MNC) = R(XRCF,MNC)(O STF (XRCF ) -OMNC (XRCF )) (97)
and
R(STF,MNC) = R(XRCF,MNC)R(FAM,XRCF )R(STF, SIC) (98)
(The earlier versions of this memo used a misalignment matrix to allow for FAM axis misalignment
with the XRCF. In the present version it is assumed that this misalignment has been taken out by
Ball in the data coming from the FAM).
Now
O STF (XRCF ) = O FAM (XRCF ) + dO FAM (XRCF ) + P FAM (XRCF ) (99)
where O FAM is the XRCF focus at XRCF coordinates (XH­f,0,0) and dO is the o#set of the FAM
coordinate origin relative to that point. The trouble is that we don't reliably know dO FAM and it
57

can change every time the FAM is switched o# and on. It must be derived from measurements of
the beam centroid. The FAM rotation matrix is
R(FAM,XRCF ) = Rot(3#/2, # x , #/2)Rot(# y , # z , 0) (100)
12 AXAF Spacecraft
In this section I describe the geometry of the AXAF spacecraft and other coordinate systems used
in the project.
. The primary positional reference is the HRMA Central Aperture Plate (CAP) reference point,
which I will designate H1. This is the origin of AXAF Project HRMA coordinates. It is
o#set from our data analysis reference point, the nominal optical node H0. H1 is a physical
datum which can be surveyed, while H0 is a theoretical point which must be modelled.
It is the nominal `thin lens' position from which we measure o#­axis angles at the focal
plane ­ it's where the photons `appear to come from'. If this were exactly true, we'd have a
constant plate scale, but actually the e#ective nodal point depends both on energy and o#­axis
angle. Nevertheless, we can adopt a conventional nodal point and make astrometric position
corrections as a function of position and energy, relative to the positions derived using that
conventional nodal point.
. Another reference point is G1, the origin of project OTG coordinates. The analogous data
analysis reference point is G0, the nominal grating node.
. Reference point X1 marks the origin of Spacecraft Coordinates, a fictitious point out in space
beyond the SIM.
. Reference point F is the flight focal position.
. A0, the center of the aperture in the spacecraft­to­IUS interface plane.
. A1, the center of the aperture at the front edge of the paraboloids (which is the origin of the
ray trace system).
12.1 Project coordinate systems
12.1.1 Spacecraft coordinates (SC­1.0)
The Spacecraft (Observatory) coordinate system XA , YA , ZA (SC/TS ICD 4 Nov 1992, 3.2.1.1.1) is
as follows: [1],[2] The X axis is parallel to the HRMA centerline; the aperture is toward positive
X, while the SIM is at lower values of X. The Y and Z axes are in the plane of the SIM, such that
58

the ACIS radiator is toward positive Z and the Y axes completes an X, Y, Z right handed set. The
coordinates are also called SCX, SCY, SCZ when we want to refer to a spacecraft coordinate system
in general rather than AXAF in particular. (XA , YA , ZA ) are measured in `inches', which is a unit
of length defined to be exactly 0.02540m. The center of the aperture in the AXAF/IUS interface
plane is defined to have coordinates (500, 0, 0). The spacecraft trunnion plane has SCZ=14. The
coordinates of the HRMA nodal point H0 in SC coordinates are (XA (H0), 0, 0) where the value of
XA (H0) is 428.116. The focus is at (31.836, 0.0, 0.0).
12.1.2 HRMA coordinates (HRMA­2.0)
The project has defined Cartesian coordinates (XH , YH , ZH ) fixed in the HRMA cap midplane,
which are called HRMA coordinates. The origin of HRMA coordinates H1 is at the HRMA
CAP reference point. The HRMA XH axis is along the HRMA optical axis, and positive X is
toward the entrance aperture. Units of HRMA coordinates shall be mm.
The Spacecraft and HRMA coordinate systems are the fundamental ones for hardware alignment,
but for data analysis of observations it's easier to use the HRMA nodal point H0 as origin. This
point is calculated to be o#set by 18.15 mm from the HRMA CAP datum A. The HRMA mirror
nodal coordinate (MNC) system is (XN , YN , ZN ), where
# # #
XN
YN
ZN
# # # =
# # #
XH -XH (H0)
YH
ZH
# # # (101)
The finite conjugate focus is at HRMA nodal coordinates (­10256.0, 0.0, 0.0). (IF1­20 Obs/SI ICD).
Note that
# # #
XN
YN
ZN
# # # =
# # #
(SCX - SCX(H0)) # 25.4
SCY # 25.4
SCZ # 25.4
# # # (102)
where SCX(H0) (or XA (H0)) is +428.116.
In the following table are listed reference points for various parts of the spacecraft. For XRCF,
I give XRCF coordinates in inches for easy comparison with the drawings, although elsewhere in
this document XRCF coordinates are measured in mm. The original interface document gave a
di#erent position for the HRMA node from the one currently in use, so these numbers need to be
rechecked.
Table 32: Interesting points in spacecraft and HRMA
nodal coordinates
Flight
Point Description (XA , YA , ZA ) (XN , YN , ZN ) Ref
59

X1 SC coordinate origin (0.0, 0.0, 0.0) (­10874.146, 0.0, 0.0)
F Flight Focus (31.836, 0.0, 0.0) (­10061.0, 0.0, 0.0) [1],p.9
Translation Table surface (55.836, 0.0, 0.0) (­9455.91, 0.0, 0.0) [1],p.17
ISIM to OBA interface (60.336, 0.0, 0.0) (­9341.6, 0.0, 0.0) [1],p.9
G1 OTG Origin (369.245, 0.0, 0.0) (­1495.3, 0.0, 0.0) [1],dr. 301331/3
OTG Datum (370.915, 0.0, 0.0) (­1452.9, 0.0, 0.0) [1],dr. 301331/3
G0 OTG Node (371.745, 0.0, 0.0) (­1431.810, 0.0, 0.0) [1], dr.301331/3
H1 HRMA CAP reference point (427.745, 0.0, 0.0) (­9.42, 0.0, 0.0) [1], dr.301331/3
H1 HRMA CAP Datum A (18.15, 0.0, 0.0) [?]
H0 HRMA nodal point (428.116, 0.0, 0.0) (0.0, 0.0, 0.0) [1], dr.30133/31
A1 Front end of paraboloids (462.474, 0.0, 0.0) (872.692, 0.0, 0.0)
A0 Aperture center in IUS I/F
plane
(500.0, 0.0, 0.0) (1825.85, 0.0, 0.0) [2],[1],p.9
XRCF
Point Description XXRCF (in) (XN , YN , ZN ) Ref
F0 XRCF Default Focus ­403.5 (­10258.3, 0.0, 0.0) [1], dr. 301331/5
G1 OTG Origin at XRCF ­60.613 (­1549.0, 0.0, 0.0) [1]
OTG Datum at XRCF ­58.943 (­1506.57, 0.0, 0.0) [1]
G0 OTG Node at XRCF ­58.113 (­1485.49, 0.0, 0.0) [1]
H1 HRMA CAP reference point 0.0 (­9.42, 0.0, 0.0)
H0 HRMA nodal point ­0.37 (0.0, 0.0, 0.0 )
Table 33: Geometrical layout of mirrors and detectors:
Value of MNC X Coord XN
Point Description TMA/TOGA HRMA (XRCF) HRMA (Flight)
F (old) On Axis Focus ­6068.61 ­10258.3 ­100065.5
F On Axis Focus ­6068.61 ­10256.0 ­10061.0
G0 On Axis Rowland Circle ­702.06 ­1485.49 ­1431.81
Shutter Plane ­546.06 Unknown None
Hyperbola back ­400.88 Unknown Unknown
Hyperbola front ­40.43 Unknown Unknown
H0 Mirror Node 0.0 0.0 0.0
Parabola back 40.42 Unknown Unknown
A1 Front end of mirror 400.94 872.692 872.692
60

Table 34: Mirror radii, mm
Description TMA/TOGA Shell 1 Shell 3 Shell 4 Shell 6
Gratings 187.7?
Hyperbola back 199.87773
Hyperbola front 209.07069
Midplane 210.00000
Parabola back 210.30940
Parabola front 213.33744 1200 960 850 620
12.2 Project Coordinate Systems
There are a plethora of existing coordinate systems in use describing positions relative to the HRMA
mirror. They are used in the assembly and alignment of the hardware but they will not be used in
the kind of data analysis I am concentrating on in this memo. They are presented for reference.
12.2.1 Orbiter coordinate system
The Space Shuttle Orbiter coordinate system [1] is used to locate the spacecraft within the Orbiter
payload bay during the launch phase. It is measured in inches with the +XO axis downwards at
launch (i.e. toward orbiter aft end), the +YO axis to starboard, and the +ZO axis toward the top
side (tail side) of the orbiter (i.e. upwards at landing). During launch, the origin of spacecraft
coordinates is at orbiter coordinates
# # #
XO
YO
ZO
# # # =
# # #
596.0
0.0
400.0
# # # (103)
The relation between Orbiter and Spacecraft coordinates is
# # #
XA
YA
ZA
# # # =
# # #
XO - 596.0
YO
ZO - 400.0
# # # (104)
12.2.2 Payload coordinate system
The Space Shuttle payload coordinate system [1] is fixed with respect to the payload, probably. It
is measured in inches. In the stowed position, payload and Orbiter coordinates are parallel, and
related by
# # #
X P
Y P
Z P
# # # =
# # #
XO - 596.0
YO
ZO - 200.0
# # # =
# # #
XA
YA
ZA + 200.0
# # # (105)
61

(reference [1].)
12.2.3 The Telescope Ensemble Coordinate System
The document EQ7­002 Rev D, [4] describing the HRMA, defines the Telescope Ensemble Coor­
dinate System (X T , Y T , Z T ) with somewhat di#erent axis choice (+Z is the optical axis) and with
origin at the focus. Presumably this is the on­orbit, zero­g focus f0, but the document doesn't say.
The document IF1­20 OBS/SI ICD [1] defines the Telescope Coordinate System (X T , Y T , Z T ) to be
identical with spacecraft coordinates.
12.2.4 Optical Bench Assembly system
The OBA system [2] is
# # #
XOBA
YOBA
ZOBA
# # # =
# # #
X FP - 28.5
Y FP
Z FP
# # # =
# # #
XA - 60.336
YA
ZA
# # # (106)
(TRW AXAF I system alignment plan). I won't discuss it further.
12.2.5 OTG Coordinates (OTG­2.0)
The OTG coordinates system (XG , YG , ZG ) has XG parallel to XH and YG , ZG parallel to YA , ZA
XG = 0 is the side of the OTG closest to the focal plane. This system (AXAF­OTG­2.0) is o#set
by 2.50 in (63.5mm) relative to the Grating Nodal system (AXAF­OTG­1.0) of interest for data
analysis.
The HETG dispersion direction is rotated 5 deg counterclockwise from YA viewed from XA = 0.
METG is rotated the same amount clockwise. The inner and outer radii of the OTG are 234.95
and 558.80 mm.
12.2.6 Project FPSI Coordinate System
The Focal Plane Science Instruments coordinate system (X F , Y F , Z F ) is essentially identical to the
LSI system defined in this document. However, the SE30 [2] definition specifies the origin as the
`desired aim point', while my LSI definition specifically selects a single nominal aim point for each
instrument.
12.2.7 SIM and ISIM coordinates
ISIM coordinates are defined in the OBS/SI ICD [1] with their origin at spacecraft coordinates
(31.836, 0.0, 0.0) near the focus (and thus defined only while the spacecraft is assembled); SIM
coordinates are defined in the System Alignment Plan D17388, with their origin at the SIM/OBA
62

interface. Both are usually measured in inches. I recommend use of the more generally defined data
analysis coordinate systems FP (fixed wrt the HRMA) and STF (fixed wrt the SIM) defined below.
SIM and OBA coordinates are identical after assembly.
(X ISIM , Y ISIM , Z ISIM ) = (XA , YA , ZA ) -
# # #
31.836
0
0
# # #
and
(X SIM , Y SIM , Z SIM ) = (XA , YA , ZA ) -
# # #
60.336
0
0
# # # .
12.2.8 SAOSAC coordinates (HRMA­3.1)
The SAOSAC coordinate system (XOSAC , YOSAC , ZOSAC ) is slightly di#erent again. In the SAOSAC
system, at XRCF, the Z­axis increases towards the SI along the FOA, while the X and Y axes are
in the aperture plane with the +Y axis vertical. The origin of SAOSAC coordinates is at CAP
Datum A, which has nodal X­coordinate XN (DatumA) = 18.15.
# # #
XOSAC
YOSAC
ZOSAC
# # # =
# # #
YN
-ZN XN (A) -XN
# # # (107)
12.2.9 Summary of useful HRMA Cartesian systems
# # #
XN
YN
ZN
# # # =
# # #
XH -XH (H0)
YH
ZH
# # # =
# # #
XN (A1) - ZOSAC
+XOSAC
-YOSAC
# # # (108)
=
# # #
(SCX - SCX(H0)) # 25.4
SCY # 25.4
SCZ # 25.4
# # #
References
[1] TRW IF1­20, Observatory to Science Instrument ICD, 11 Jan 1996.
[2] TRW SE30, AXAF­I System Alignment Plan, D17388, 11 Jan 1996.
[3] AXAF Transmission Grating Di#raction Coordinates, ASC Memo, D.P. Huenemorder, 2 Apr
1996.
63

[4] EQ7­002 Rev D.
64

Figure 16: The AXAF SIM Translation Table, showing the flight focal plane instruments to scale.
Distances are in mm. Coordinate system is AXAF­STT­1.0.
65

STT
HRC
ACIS
SIM
HRMA
H0
Z
Y
Z STF
Y STF
X STF
X N
N
N
Figure 17: HRMA Nodal and STF Coordinates showing the on­orbit configuration. The SIM
Transfer Table (STT) carrying the instruments moves along the Z STF axis to select an instrument
and along the X STF axis to adjust focus.
H0
f
q
X H
Figure 18: HRMA spherical coordinates
H0
X H
HRMA equator
HRMA meridian
az=0
el=0
Figure 19: HRMA source coordinates
66

Default Configuration C0:
HRMA, SI along FOA
HRMA SI FAM
H0
X
Z
Y XRCF
XRCF
XRCF
S
Figure 20: XRCF Coordinates showing the general configuration with HRMA, DFC and LSI coor­
dinate systems
HRMA SI
H0
S0
X
Z
Y
XRCF
XRCF
XRCF
X
Z
Y
H
H
H
X DFC
Y DFC
Z
DFC
Z LSI
LSI
X
Y LSI
SIM
FAM
Figure 21: XRCF Coordinates showing the general configuration with HRMA and LSI coordinate
systems
67

A0 A1 H0 H1 OTG
ISIM/
OBA
F
SCX
SCZ
SCY
G
X1
Figure 22: Schematic of interesting points in the spacecraft
68