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# Lecture 20 - PowerPoint PPT Presentation

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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• A couple of quick additions to past topics:

• WCS keywords of FITS files

• Back to XMM calibration

• hardness ratios

• photon index vs energy index

• 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.

WCS must encode the

relation between θ and the

pixel number.

Pixel grid on tangent plane

θ

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.

Note:

(1) pref can be real-valued;

(2) By convention, p at the centre

of the 1st pixel = 1.0.

• 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.

• 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

Silicon K edge

Oxygen K edge

Gold M edge

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.

dimensionless

erg s-1 eV-1 cm-2

cm2

photons s-1 eV-1

 E of course is the photon energy.

• A reasonable breakdown of AX is

where

• R is the redistribution matrix;

• A is the on-axis effective area (including filter and QE contributions);

• V is the vignetting function;

• 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:

• 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

photons

erg s-1 eV-1 cm-2

s

• 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

• 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’.

• 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.

• 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.

• 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.

Total energy S of all the N photons in a bin

of centre energy E is (about) N times E.

Photon energy

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

• Hence, if

then

•  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!