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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

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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.
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!