1. NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 7 Confusion Dynamic range Resolved sources Selection biases Luminosity (and mass) functions Volume- vs flux-limited surveys.

2. NASSP Masters 5003F - Computational Astronomy - 2009 A 1D simulation. Start with a distribution of sources. Euclidean model gives: Each source has some random structure. They also vary in width. n(S) α S -5/2 (Actually I used a lower power to make the plots look better.)

3. NASSP Masters 5003F - Computational Astronomy - 2009 A 1D simulation. Add in instrumental broadening.

4. NASSP Masters 5003F - Computational Astronomy - 2009 A 1D simulation. And finally, add noise. (Remember, it can happen the other way around – first noise then broadening.) Sensitivity here is limited by noise. Suppose we push the noise right down, by observing longer, or with a more sensitive instrument…?

5. NASSP Masters 5003F - Computational Astronomy - 2009 Confusion …Eventually the sensitivity becomes confusion-limited. At each point in the sky, the nett flux is a sum of contributions from >1 source. –Brightest contributor named the confused source; its flux and position are distorted. –All fainter are not directly observable. –But, can get statistical info on n(S) from noise distribution.

6. NASSP Masters 5003F - Computational Astronomy - 2009 Near-confused fields: A NICMOS exposure towards the galactic centre. Credit: Spitzer Science Centre/STScI An all-instrument mosaic of XMM EPIC cameras. A rich stellar cluster.

7. NASSP Masters 5003F - Computational Astronomy - 2009 Two possible remedies: 1.Subtract sources, starting with the brightest. –Eg the CLEAN algorithm in radio interferometry. –Eg 2: sExtractor. Brightest subtracted

8. NASSP Masters 5003F - Computational Astronomy - 2009 Dynamic range The problem with this is that the subtraction may not be perfect. –Imperfect measurement of source position or flux. –Calibration errors (interferometry). –Imperfect knowledge of the source profile (XMM). Ratio of brightest source to remaining artifacts called the dynamic range. Imperfect subtraction of the PSF in a MERLIN image. Best dynamic range only 10 4 (=40 dB).

9. NASSP Masters 5003F - Computational Astronomy - 2009 Example: recently discovered exoplanets Planet 1 Planet 2 Credit: Gemini Observatory/AURA

10. NASSP Masters 5003F - Computational Astronomy - 2009 Example: group (cluster?) with Cen B Smoothed Dummy Raw Bright sources subtracted Schroeder, Mamon and Stewart – in preparation.

11. NASSP Masters 5003F - Computational Astronomy - 2009 The other possible remedy: 2.Try to reduce the instrumental broadening.

12. NASSP Masters 5003F - Computational Astronomy - 2009 Higher resolution Methods: –Via hardware: wider aperture – higher resolution. –Or software: deconvolution (eg Maximum Entropy). The fundamental limit comes from the widths of the objects themselves – ‘natural confusion.’

13. NASSP Masters 5003F - Computational Astronomy - 2009 Eg the Hubble Deep Field. Credit STScI

14. NASSP Masters 5003F - Computational Astronomy - 2009 Detecting resolved sources. Our earlier assumption that we knew the form of S is no longer true. Some solutions: 1.Combine results of several filterings. (Crudely done in XMM.) But, ‘space’ of possible shapes is large. Difficult to calculate nett sensitivity. 2.Wavelet methods.

15. NASSP Masters 5003F - Computational Astronomy - 2009 Wavelet example Raw dataWavelet smoothed F Damiani et al (1997) Multi-scale wavelets can be chosen to return best-fit ellipsoids.

16. NASSP Masters 5003F - Computational Astronomy - 2009 Selection biases Fundamental aim of most surveys is to obtain measurements of an ‘unbiased sample’ of a type of object. Selection bias happens when the survey is more sensitive to some classes of source than others. –Eg, intrinsically brighter sources, obviously. Problem is even greater for resolved sources. –Note: ‘resolved’ does not just mean in spatial terms. Eg XMM or (single-dish HI surveys) in which most sources are unresolved spatially, but well resolved spectrally.

17. NASSP Masters 5003F - Computational Astronomy - 2009 Examples Optical surveys of galaxies. Easiest detected are: –The brightest (highest apparent magnitude). –Edge-on spirals. HI (ie, 21 cm radio) surveys of galaxies. Easiest detected are: –Those with most HI mass (excludes ellipticals). –Those which don’t ‘fill the beam’ (ie are unresolved). Note: where sources are resolved, detection sensitivity tends to depend more on surface brightness than total flux.

18. NASSP Masters 5003F - Computational Astronomy - 2009 Full spatial information Q: We have a low-flux source - how do we tell whether it is a high-luminosity but distant object, or a low-luminosity nearby one? A: Various distance measures. –Parallax - only for nearby stars – but Gaia will change that. –Special knowledge which lets us estimate luminosity (eg Herzsprung-Russell diagram). –Redshift => distance via the Hubble relation. This is probably the most widely used method for extragalactic objects.

19. NASSP Masters 5003F - Computational Astronomy - 2009 Luminosity function Frequency distribution of luminosity (luminosity = intrinsic brightness). The faint end is the hardest to determine. –Stars – how many brown dwarfs? –Galaxies – how many dwarfs? Distribution for most objects has a long faint-end ‘tail’. –Schechter functions. P Kroupa (1995) P Schechter (1976)

20. NASSP Masters 5003F - Computational Astronomy - 2009 HI mass function Red shift is directly measured. Flux is proportional to mass of neutral hydrogen (HI). –Hence: usual to talk about HI mass function rather than luminosity function. S E Schneider (1996)

21. NASSP Masters 5003F - Computational Astronomy - 2009 Relation to logN-logS Just as flux S is related to luminosity L and distance D by So is the logN-logS – or, to be more exact, the number density as a function of flux, n(S) - a convolution between the luminosity function n(L) and the true spatial distribution n(D). BUT… –The luminosity function can change with age – that is, with distance! (And with environment.) S α L/D2S α L/D2

22. NASSP Masters 5003F - Computational Astronomy - 2009 Volume- vs flux-limited surveys Information about the distance of sources allows one to set a distance cutoff, within which one estimates the survey is reasonably complete (ie, nearly all the available sources are detected). Such a survey is called volume-limited. It allows the luminosity (or mass) function to be estimated without significant bias. –However, there may be few bright sources. Allow everything in, and you have a flux- limited survey. –Many more sources => better stats; but biased (Malmquist bias).