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Coordinate Systems for Analysis of On­Orbit
Chandra Data
Paper III: Dispersed Spectra
Jonathan McDowell
Harvard­Smithsonian Center for Astrophysics
David Huenemorder
MIT Center for Space Research
1999 Nov 29
1 Introduction
The Chandra X­ray Observatory, launched in July 1999, provides X­ray imaging and spectral data
of unprecedented resolution. The geometry of Chandra's detectors is more complicated than those
of previous missions; this, coupled with the higher accuracy requirements, means that more care
must be taken to derive accurate coordinates and to distinguish clearly between different coordinate
systems.
In the first two papers in this series we discussed calculation of image space coordinates. In the
present note, we consider interpretation of grating data.
1.1 Chandra gratings
The Chandra X­ray Observatory has two transmission gratings, the HETG (High Energy Transmis­
sion Grating) and the LETG (Low Energy Transmission Grating). Each of these is physically a large
circular structure consisting of four annuli matched to the radii of the four HRMA (High Resolution
Mirror Array) mirror pairs. Around the annuli are mounted hundreds of individual grating facets.
The grating structures are attached to the back of the HRMA with hinges so that either of them may
be rotated around to intercept the rays emerging from the mirrors. The LETG facets, collectively
the LEG (Low Energy Grating), have a period of around 9900 š A and are aligned so that the light
is dispersed along the telescope FCY (paper I) axis, perpendicular to the direction of optical bench
motion and aligned with the long direction of the HRC­S and ACIS­S detectors. The HETG has
two sets of facets, known as HEG (High Energy Grating) and MEG (Medium Energy Grating), with
periods of 2000 and 4000 š A respectively. The HEG and MEG dispersion directions are each inclined
about 5 degrees to the FCY axis so that the dispersed spectra form an X shape on the imaging
detectors. In normal use, the LETG is employed in conjunction with the HRC­S detector, since
ACIS doesn't have good response at low energy. The HETG is employed with the ACIS­S detector,
whose independent CCD energy resolution facilitates order separation.
In the idealization we adopt here, light from the HRMA passes through a single point G, the
grating node, from which it is dispersed by an angle ` along the grating dispersion direction. We
refer to the LEG, HEG and MEG as `grating arms' and consider only one grating arm at a time.
1

The distance between G and the focus F is the Rowland distance R, and the best focus for the
dispersed photons lies along a circle in the dispersion plane centered on G with radius R.
A photon of wavelength – passing through G will be deflected by an angle
r = sin \Gamma1 (m–=P )
where P is the `grating period' and m, an integer, is the `order'. Most photons enter the zero order
spectrum with m=0, and land exactly where they would have if the grating had not been present ­
the zero order position ZO. The next most probable fate for the photons is to be deflected into the
m=1 or m=­1 first order spectra; successively higher order spectra contain less and less of the total
incident energy.
2 Source based coordinate systems
To analyse the geometry of a spectrum we need to set up coordinate systems for a particular source.
Note that if multiple sources are present in the field, their spectra will overlap on the detector making
analysis difficult.
Consider a sphere whose equatorial plane lies in the dispersion direction and whose center is
at G. Then the dispersion angle r is just longitude on the sphere relative to a meridian passing
through the line S joining G with ZO, the source's zero order position. The latitude d is called the
cross­dispersion angle.
If the source is on axis, ZO lies at the imaging aimpoint and in the mirror nodal coordinates of
paper I the pole of the sphere has coordinates
d 0 = (0; \Gamma sin ff G ; cos ff G ) (1)
where ff G is the angle between the dispersion direction and the spacecraft Y axis. For an off­axis
source, the dispersion direction is perpendicular to both d 0 and S; we define a cartesian orthonormal
set of Grating Zero Order coordinates (GZO)
eXZO = \GammaS=jSj
e YZO = d 0 “ eXZO =jd 0 “ eXZO j
eZZO = eXZO “ e YZO
(2)
with origin at G. Diffracted photons travel in the XZO ; YZO plane, and the intersection of this plane
with the detector surface defines the dispersion direction.
To calculate the wavelength of a photon, we go through several steps:
1. We use the equations of paper I to convert from detected chip coordinates to focal coordinates
(FCX,FCY,FCZ), correcting for the chip position on the SIM and for the SIM translation
and boresight. We now have the position of the dispersed photon relative to the mirror axis
(assumed to coincide with the grating axis).
2. We then calculate the GZO coordinates of the photon by subtracting the FC coordinates of
G (which are just (R,0,0)) and applying a rotation matrix to account for the misalignment of
the GZO axes with the FC axes.
0
@
XGZO
YGZO
ZGZO
1
A = R(FC;GZO)
0
@
0
@
XFC
YFC
ZFC
1
A \Gamma
0
@
R
0
0
1
A
1
A
2

d 0
ZO
Y
X
Z
Z ZO
X ZO
Y ZO
Y
ZO
ZO
G
F
Z
Figure 1: Grating Zero Order coordinates
3. We now convert from GZO coordinates to longitude and latitude on the GZO sphere,
r = tan \Gamma1
i
\GammaY GZO
XGZO
j
¸ \GammaY GZO=XGZO
d = tan \Gamma1
`
+ZGZO
p
X 2
GZO +Y 2
GZO
'
¸ +ZGZO=XGZO
(3)
4. Finally, we apply the dispersion relation
– = P sin r=m
to derive the wavelength assuming the order m is known.
2.1 Grating Diffraction Plane Pixel Coordinates (GDP­1.1)
For analysis purposes it may be useful to define a standard pixel system for creating grating images.
The Grating Diffraction Plane Pixel Coordinates GDX, GDY are defined by
GDX = GDX0 \Gamma \Delta \Gamma1
gs (Y ZO=XZO )
GDY = GDY 0 + \Delta \Gamma1
gs (ZZO=XZO ) (4)
analogously to the Focal Plane Pixel Coordinates. The pixel size \Delta gs is chosen to match the
instrument physical pixel size (and in the current GDP­1.1 system is set to 0.154 arcsec, with GDX0
= GDY0 = 32768.50 pixels).
They are related to the angular Grating Diffraction Coordinates by
GDX = GDX0+ \Delta \Gamma1
gs tan r
GDY = GDY 0 + \Delta \Gamma1
gs tan d cos r
(5)
3 In­flight numerical values for Chandra
The Rowland diameter is 8632.48mm.
Table 1: Grating properties
Instrument P ff G (deg)
3

HETG 2000.81 š A ­5.18
METG 4001.41 š A 4.75
LETG 9912.16 š A +0.016
4 Summary
In this paper we summarized the calculations used by the Chandra data processing system to assign
wavelengths to photons in transmission grating data. The formalism is sufficiently general that it
should be applicable to other missions by appropriately changing the numerical values in section 3.
We thank Dave Huenemorder and Dan Dewey for useful discussions, and Helen He for implement­
ing the software routines used to apply the coordinate transforms in the Chandra data processing
system.
4

Contents
1 Introduction 1
1.1 Chandra gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Source based coordinate systems 2
2.1 Grating Diffraction Plane Pixel Coordinates (GDP­1.1) . . . . . . . . . . . . . . . . 3
3 In­flight numerical values for Chandra 3
4 Summary 4
List of Tables
1 Grating properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
List of Figures
1 Grating Zero Order coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
5