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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. WCS keywords of FITS files. WCS stands for W orld C oordinate S ystem. http://fits.gsfc.nasa.gov/fits_wcs.html

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Lecture 20
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

Wcs keywords of fits files
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.

Eg projection onto a tangent plane
Eg, projection onto a tangent plane.

WCS must encode the

relation between θ and the

pixel number.

Pixel grid on tangent plane


Wcs example continued
WCS example continued

w- wref

The general formula in this case is

(p - pref) * scale = tan(w – wref).

p is the pixel coordinate and w the

world coordinate. w might eg be

right ascension or declination.

p- pref

  • WCS must describe 4 things:

  • pref

  • wref

  • scale

  • the nature of the functional relation.

  • Perhaps also world units.


(1) pref can be real-valued;

(2) By convention, p at the centre

of the 1st pixel = 1.0.

Wcs keywords for array extensions
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 addingPCi_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.

Now a little word more about aips
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.

Back to xmm calibration quantities 1 quantum efficiency
Back to XMM.Calibration quantities: (1) Quantum Efficiency

Silicon K edge

Oxygen K edge

Effective area change with filters
Effective Area change with filters

This is for pn – MOS is very similar.


  • Relation between incident flux density S and the photon flux density φ: most general form is

    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.

    • 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.


erg s-1 eV-1 cm-2


photons s-1 eV-1

 E of course is the photon energy.


  • A reasonable breakdown of AX is


    • 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.

All dimensionless except A.


  • 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

    This gets rid of the integral over r, and the r‘ in V and ρ become r0.


  • What we do next depends on the sort of product which we want. There are really only 4 types (XMM pipeline products) to consider:

Exposure map
Exposure map

  • For XMM images we have

    where the exposure mapε is

    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.

photons cm2

eV s-1 erg-1


erg s-1 eV-1 cm-2



  • For XMM spectra

    where the ancillary response function (ARF) α is

    This is a bit more rigorous because the resulting spectrum q is explicitly acknowledged to be pre-RM.

    • And if S can be taken to be time-invariant, then this expression follows almost exactly from the general expression involving X.

photons eV-1

Fractional exposure
Fractional exposure

  • For XMM light curves,

    where the fractional exposuref is

photons s-1


  • There is just a small modification to the ‘image’ approximation:

    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’.

Prescriptions to obtain ergs s 1
Prescriptions to obtain ergs s-1:

  • Image:

    • Divide by exposure map

    • Multiply by ECF

  • Spectrum:

    • You don’t. Compare to forward-treated model instead.

  • Light curve:

    • Divide by frac exp

    • Multiply by ECF

  • Source:

    • As for image but also divide by integral of ρC.

Some spectral lore 1 hardness ratios
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

  • 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.

Some spectral lore 2 photon index
Some spectral lore: (2) Photon index.

  • Suppose a source has a power spectrum, ie

  • 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.

Photon frequency histogram
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

Photon index
Photon index.

  • Thus the energy spectrum S(E) and the photon spectrum N(E) are related by

  • Hence, if


  •  photon index is always 1 less than the spectral index.

Matters aren’t helped by

the habit to use eV for the

photon energy but ergs

for the total energy!