Lecture 20

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

2. 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.

3. Eg, projection onto a tangent plane. WCS must encode the relation between θ and the pixel number. Pixel grid on tangent plane θ

4. 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.

5. 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.

6. 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.

7. Back to XMM.Calibration quantities: (1) Quantum Efficiency Silicon K edge Oxygen K edge

8. (2) Effective area (no filter) (includes QE) Gold M edge

9. Effective Area change with filters This is for pn – MOS is very similar.

10. 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.

11. 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.

12. 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.

13. 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:

14. 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

15. 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

16. Fractional exposure • For XMM light curves, where the fractional exposuref is photons s-1

17. 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’.

18. 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.

19. 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.

20. 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.

21. 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

22. 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!