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## Lecture 20

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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 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.

WCS must encode the

relation between θ and the

pixel number.

Pixel grid on tangent plane

θ

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.

Note:

(1) pref can be real-valued;

(2) By convention, p at the centre

of the 1st pixel = 1.0.

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.

- 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.

(2) Effective area (no filter) (includes QE)

Gold M edge

Effective Area change with filters

This is for pn – MOS is very similar.

Exposure

- 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.

Exposure

- 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;
- 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.

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

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

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:

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

photons

erg s-1 eV-1 cm-2

s

ARF

- 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

Sources

- 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:

- 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.

- 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.

- 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

Total energy S of all the N photons in a bin

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

Photon energy

Photon index.

- 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!

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