Transcript Lecture 1

Lecture 20
• A couple of quick additions to past topics:
– WCS keywords of FITS files
– Common AIPS tasks
• Back to XMM calibration
• hardness ratios
• photon index vs energy index
NASSP Masters 5003F - Computational Astronomy - 2009
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WCS keywords of FITS files
WCS stands for World Coordinate System.
http://fits.gsfc.nasa.gov/fits_wcs.html
What they’re for: pixellated data – ie
samples of some quantity on a regular
grid.
WCS keywords define the mapping
between the pixel index and a world
coordinate system.
Eg: a 2d image of the sky. We want to
know which sky direction the (j,k)th pixel
corresponds to.
NASSP Masters 5003F - Computational Astronomy - 2009
Eg, projection onto a tangent plane.
WCS must encode the
relation between θ and the
pixel number.
θ
NASSP Masters 5003F - Computational Astronomy - 2009
WCS example continued
The general formula in this case is
w- wref
(p - pref) * scale = tan(w – wref).
p is the pixel coordinate and w the
world coordinate. w might eg be
right ascension or declination.
WCS must describe 4 things:
1. pref
2. wref
3. scale
4. the nature of the functional relation.
Perhaps also world units.
p- pref
Note:
(1) pref can be real-valued;
(2) By convention, p at the centre
of the 1st pixel = 1.0.
NASSP Masters 5003F - Computational Astronomy - 2009
WCS keywords for array extensions
• In what follows, n is an integer, corresponding to
one of the dimensions of the array.
–
–
–
–
CRVALn – wref.
CRPIXn – pref.
CDELTn – scale.
CTYPEn – an 8-character string encoding the function
type (eg ‘TAN---RA’). There is an agreed list of these.
– CUNITn – string encoding the unit of w (eg ‘deg’). Also an
agreed list.
• In addition, rotated coordinate systems can be
defined via either adding PCi_j keywords to the
above scheme, or replacing CDELTn by CDi_j
keywords. But I don’t want to get too deeply into
this.
• Analogous (starting with T) WCS keywords are also defined for table
columns.
NASSP Masters 5003F - Computational Astronomy - 2009
Now... a little word more about AIPS.
• If you look at the cookbook, you will see there are
hundreds of AIPS tasks. It is a bit daunting.
• However, you will probably only ever use the
following:
–
–
–
–
FITLD – to import your data from FITS.
IBLED – lets you flag bad data.
CALIB – to calculate calibration tables.
SPLIT – splits your starting single observation file into 1
UV dataset per source.
• Usually you will observe 3 or maybe 4 sources during your
observation – the target, a primary and secondary flux calibrator
and a phase calibrator.
– IMAGR – to calibrate, grid, FT and clean your data.
– FITAB – exports back to FITS.
NASSP Masters 5003F - Computational Astronomy - 2009
Back to XMM.
Calibration quantities: (1) Quantum Efficiency
Silicon K edge
Oxygen K edge
NASSP Masters 5003F - Computational Astronomy - 2009
(2) Effective area (no filter) (includes QE)
Gold M edge
NASSP Masters 5003F - Computational Astronomy - 2009
Effective Area change with filters
This is for pn – MOS is very similar.
NASSP Masters 5003F - Computational Astronomy - 2009
Exposure
• Relation between incident flux density S
and the photon flux density φ: most
general form is
dimensionless
erg s eV cm
cm
photons s eV
dE 
 E , r, t   A
dr X E , E , r, r, t  S E , r, t 
E   E of course is the photon energy.
where A is an effective area and the
fractional exposure kernel X contains all
the information about how the photon
properties are attenuated and distributed.
-1
-1
2
-1
-1
-2
– Note I didn’t include a t' because in XMM
there is no redistribution (ie ‘smearing’)
mechanism which acts on the arrival time.
• Vector r is shorthand for x,y.
NASSP Masters 5003F - Computational Astronomy - 2009
Exposure
• A reasonable breakdown of AX is
AX  RE, E AEV E, rCr  r, r, EDt 
where
All dimensionless except A.
– R is the redistribution matrix;
– A is the on-axis effective area (including filter and QE
contributions);
– V is the vignetting function;
– C holds information about chip gaps and bad pixels;
– ρ is the PSF (including OOTE and RGA smearing); and
– D is a ‘dead time’ fraction, which is a product of
• a fixed fraction due to the readout cycle, and
• a time-variable fraction due to blockage by discarded cosmic
rays.
• the fraction of ‘good time’ during the observation.
NASSP Masters 5003F - Computational Astronomy - 2009
Exposure
• This includes a number of assumptions,
eg
– The spacecraft attitude is steady.
– Variations between event patterns are
ignored.
– No pileup, etc etc
• Now we try to simplify matters. First, let’s
only consider point sources, ie
S E, r, t   S E, t  r  r0 
This gets rid of the integral over r, and the r‘
in V and ρ become r0.
NASSP Masters 5003F - Computational Astronomy - 2009
•
Exposure
What we do next depends on the sort of
product which we want. There are really
only 4 types (XMM pipeline products) to
consider:
I r    dt  dE  E , r, t   an image
t
qE    dt
t
Lt  
E
 dr  E, r, t 
 a spectrum
Region
 dr  dE  E, r, t 
E
 a light curve
Region
S   dt
t
 dr  dE  E, r, t 
E
 a source
Region
NASSP Masters 5003F - Computational Astronomy - 2009
Exposure map
• For XMM images we have
photons cm
photons
erg s eV cm
eV s erg
1
I r    r  E   dt S E , r, t 
s
t t
where the exposure map ε is
2
-1
-1
-1
-1
-2
 r   V  E , r C r  dt Dt 
t
and the energy conversion factor (ECF) ψ is
calculated by integrating over a model
spectrum times R times A.
– Hmm well, it’s kind of roughly right.
NASSP Masters 5003F - Computational Astronomy - 2009
ARF
• For XMM spectra
photons eV-1
qE    E   dt
t
 dr S E, r, t 
Region
where the ancillary response function (ARF) α is
1
 E   AE V E , r0  dr C r  r0 , r, E  dt Dt 
Reg
t
E
This is a bit more rigorous because the resulting
spectrum q is explicitly acknowledged to be preRM.
– And if S can be taken to be time-invariant, then this
expression follows almost exactly from the general
expression involving X.
NASSP Masters 5003F - Computational Astronomy - 2009
Fractional exposure
• For XMM light curves,
photons s
dE
Lt   f t  A E  dr 
S E , r , t 
Reg. E E
-1
where the fractional exposure f is
f t  V  E , r0  Dt  dr C r  r0 , r, E
Reg.

NASSP Masters 5003F - Computational Astronomy - 2009
Sources
• There is just a small modification to the
‘image’ approximation:
1
S   r  E   dr  r, r, E   dt S E, r, t 
t

t
Region
This is probably the least rigorous of the
three product-specific distillations of X.
• To some extent, this idiosyncratic way of
cutting up the quantities is just ‘what the
high-energy guys are used to’.
NASSP Masters 5003F - Computational Astronomy - 2009
•
Prescriptions to obtain ergs s-1:
Image:
1. Divide by exposure map
2. Multiply by ECF
•
Spectrum:
1. You don’t. Compare to forward-treated model
instead.
•
Light curve:
1. Divide by frac exp
2. Multiply by ECF
•
Source:
3. As for image but also divide by integral of ρC.
NASSP Masters 5003F - Computational Astronomy - 2009
Some spectral lore: (1) Hardness ratios.
• This is a term you will encounter often in
the high-energy world.
– Add up the counts within energy band 1 
C1;
– add up the counts in band 2  C2;
– the hardness ratio is defined as
C2  C1
HR 
C2  C1
• Clearly confined to the interval [-1,1].
• It is a crude but ready measure of the
spectral properties of the source.
• Uncertainties are often tricky to calculate.
NASSP Masters 5003F - Computational Astronomy - 2009
Some spectral lore: (2) Photon index.
• Suppose a source has a power spectrum,
ie

S E   S0 E
• As we know, α is called the spectral index.
If we plot log(S) against log(E), we get a
straight line of slope α.
• But! Think how we measure a spectrum.
We have to count photons and construct a
frequency histogram – so many within
energy bin foo, etc.
NASSP Masters 5003F - Computational Astronomy - 2009
Photon frequency histogram
Total energy S of all the N photons in a bin
of centre energy E is (about) N times E.
Photon energy
NASSP Masters 5003F - Computational Astronomy - 2009
Photon index.
• Thus the energy spectrum S(E) and the photon
spectrum N(E) are related by
S E   E  N E 
• Hence, if
S E   E 
then
N E   E 1
•  photon index is always 1 less than the
spectral index.
NASSP Masters 5003F - Computational Astronomy - 2009