1. Likelihood analysis of small diffuse sources Riccardo Rando Elisa Mosconi, Omar Tibolla DC2 Kickoff Meeting – SLAC, 1-3 March 2006

2. DC2 Kickoff, SLAC, 1-3 March 2006 2 Outline + Intro ● A simple source ● A simple analysis ● A more challenging source ● Other ideas ● Conclusions Small sources are those that cover a small fraction of the sky In this talk I will consider sources with a diameter of a few degrees Issues are the precision of diffuse response calculation within the likelihood pkg

3. DC2 Kickoff, SLAC, 1-3 March 2006 3 Simulation Let's consider a small squarish source, side 2 degrees A uniform spectrum index  =-1.6 (MapSource) Intensity ~1/6 Crab from 20 MeV up MapSource reads in map only for relative values: all pixels =1 within source boundaries, 10 -9 elsewhere Observation: 1 month Let's make things simple: it is placed at RA,DEC=0,0 : low (galactic) bkg

4. DC2 Kickoff, SLAC, 1-3 March 2006 4 MapSource From ST confluence page: <SpectrumClass name="MapSource" params="17.,2.1,$(FLUXROOT)/sources/gas_gal.fits,30.,2e5"/> ● Total photon flux from the map (#/m^2/s) = 17. ● Photon spectral index = 2.1 ● Fits file (plate-carree only) = $(FLUXROOT)/sources/gas_gal.fits ● Minimum photon energy (MeV) = 30. ● Maximum photon energy (MeV) = 2e5 File is a flat snapshot of the source Map is a projection of the sky, pixels are bins where integrated flux must be specified in ph/cm 2 /s/sr Anyhow, in the MapSource spectrum one sets the overall normalization, so only relative intensities are relevant

5. DC2 Kickoff, SLAC, 1-3 March 2006 5 FITS file RVAL1 = -179.75 CRPIX1 = 1.00 CTYPE1 = 'RA ' CUNIT1 = 'deg ' CRVAL2 = -89.75 CRPIX2 = 1.00 CTYPE2 = 'DEC ' CUNIT2 = 'deg ' CDELT1 = 0.5 / increment on x CDELT2 = 0.5 / increment on y

6. DC2 Kickoff, SLAC, 1-3 March 2006 6 Looking at the data Let's see at the data, in 0.5x0.5 deg 2 bins, RA-DEC projection Linear scale Somehow (WT, Poisson counts) one would have to obtain the image size/shape. Very simply, I cut the simulated events below 1 GeV, and here's the result 2 deg Binning 0.25 deg = no bkg Shape is pretty clear Model source map (1 on source, ~0 elsewhere) ~2400 photons in 1 month model

7. DC2 Kickoff, SLAC, 1-3 March 2006 7 Likelihood Now you know the drill... selection (from 30 MeV) livetimecube, expmap, diffresp, fit (see likelihood tutorial) Fixed extragalactic and galactic diffuse Try with both free & fixed spectral index X0fixed: Prefactor: 5104.12 +/- 123.639 Index: -1.6 Scale: 100 Npred: 2206.8 X0: Prefactor: 5212.46 +/- 243.929 Index: -1.56128 +/- 0.0169117 Scale: 100 Npred: 2426.05 This is pretty straightforward Maybe a little low? (-2.5  ) What about this?? (units: 10 -9 ph/cm 2 /s/MeV ) How does it relate to 0.03 ph/m 2 /s from 20 MeV?

8. DC2 Kickoff, SLAC, 1-3 March 2006 8 Making sense of prefactor We measured a prefactor of (5.2±0.2)×10 -6 ph/(cm 2 s MeV) Spectral index set to -1.56 This should be 1 if the map in the model file were a true representation of the source flux from 20 MeV! I put 1 arbitrarily in all source pixels, though. Hence, the map (2x2 deg 2, 1 ph/cm 2 /s/sr, 1.22 10 -3 ph/cm 2 /s total) is off by a factor 2.29·10 -3 Correcting the map value gives 2.29·10 -3 x 1.22·10 -3 = 2.79·10 -6 /cm 2 /s This means 2.8·10 -2 ph /m 2 /s, or 0.028 ph/m 2 /s So we recover an integral flux of (0.028±0.001) ph/(m 2 s) The true value was 0.03!! Looks OK 

9. DC2 Kickoff, SLAC, 1-3 March 2006 9 Uniformity? Assess source uniformity, in emission and spectral index?? Divided the source model into 2 2x1 deg 2 pixels: is the source uniform L/R? Do new diff.resp. and likelihood Look ~ uniform Split again: 4 pixels (UL,UR,DL,DR)... LR TOT X0left: Prefactor: 5863.79 +/- 1042.95 Index: -1.61463 +/- 0.0520546 Scale: 100 Npred: 939.53 X0right: Prefactor: 4931.39 +/- 601.263 Index: -1.5385 +/- 0.0356165 Scale: 100 Npred: 1486.85 I L =0.016 ± 0.003 ph/m 2 /s  L = -1.61 ± 0.05 I R =0.013 ± 0.002 ph/m 2 /s  R = -1.54 ± 0.04 

10. DC2 Kickoff, SLAC, 1-3 March 2006 10 More complicated Intensity spectral index vary along source All events, bins 0.5 deg Above 1 GeV, bins 0.25 deg X1: Prefactor: 7080.25 +/- 294.092 Index: -1.69109 +/- 0.0167291 Scale: 100 Npred: 2607.8 X1upright: Prefactor: 5471.25 +/- 1154.14 Index: -1.61747 +/- 0.062207 Scale: 100 Npred: 710.497 X1downright: Prefactor: 16465.4 +/- 1692.57 Index: -1.90758 +/- 0.0375423 Scale: 100 Npred: 1369.86 X1upleft: Prefactor: 2174.37 +/- 708.43 Index: -1.46277 +/- 0.0854636 Scale: 100 Npred: 236.447 X1downleft: Prefactor: 2976.89 +/- 725.928 Index: -1.47621 +/- 0.0663137 Scale: 100 Npred: 313.46

11. DC2 Kickoff, SLAC, 1-3 March 2006 11 More complicated : results I=0.005 g=1.6 I=0.01 g=1.7 I=0.015 g=1.8 I=0.01 g=1.7 I UL =0.003 ± 0.001 ph/m 2 /s  DL = -1.46 ± 0.08 I DL =0.004 ± 0.001 ph/m 2 /s  DL = -1.48 ± 0.07 I DR =0.024 ± 0.002 ph/m 2 /s  DR = -1.91 ± 0.04 I UR =0.007 ± 0.002 ph/m 2 /s  UR = -1.62 ± 0.06 Statistics is probably the limiting factor (2670 photons in source ~500 photons for DL,UR) True values measured

12. DC2 Kickoff, SLAC, 1-3 March 2006 12 Other ideas I investigated a region where galactic contribution is negligible When diffuse bkg is small/negligible another strategy can be useful to gain insight on the source characteristics before trying to model it = Select events from a certain energy up, to limit effect of PSF Make a countmap, with n B bins in RA & DEC, n E bins in energy Make exposure map with identical binning in RA,DEC,energy (Several energy bins are required to have a good approx.) Divide fits files bin by bin using, e.g., farith Map of “true flux” is obtained Select regions, plot log(Ng) vs log(E) Fit and obtain spectral index I found values within 5-10% of likelihood fit 1 3 2 1 2 3 /

13. DC2 Kickoff, SLAC, 1-3 March 2006 13 Conclusions ● Likelihood seems to work adequately with small diffuse sources ● Sources can be examined and compared with a given model ● Some more advanced investigation is feasible ● Still to assess impact of diffuse backgrounds

14. DC2 Kickoff, SLAC, 1-3 March 2006 14 Troubles with obsSim There seems to be a slight inconsistency on how obsSim and likelihood read the input fits file: MapSource was defined so that pixels fall at 179.5―180.0, 180.0―180.5... Apparently likelihood does this, but obsSim adds an offset I take the “real” data for the diff source only and split them left and right, to count photons. Compare with npred Offset on the DEC direction seems absent 180.0