Confusion

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					                   Lecture 7
•   Confusion
•   Dynamic range
•   Resolved sources
•   Selection biases
•   Luminosity (and mass) functions
•   Volume- vs flux-limited surveys.




                                 NASSP Masters 5003F - Computational Astronomy - 2009
A 1D simulation.
        • Start with a
          distribution of
          sources. Euclidean
          model gives:
               n(S) α S-5/2
          (Actually I used a lower power
          to make the plots look better.)

        • Each source has
          some random
          structure.
        • They also vary in
          width.
                       NASSP Masters 5003F - Computational Astronomy - 2009
A 1D simulation.
        • Add in instrumental
          broadening.




                   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…?

                   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.
                NASSP Masters 5003F - Computational Astronomy - 2009
                                Near-confused fields:




A NICMOS exposure towards the                An all-instrument mosaic of XMM
       galactic centre.                     EPIC cameras. A rich stellar cluster.
     Credit: Spitzer Science Centre/STScI




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




                                 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
Imperfect subtraction of the PSF in
a MERLIN image. Best dynamic range      dynamic range.
only 104 (=40 dB).
                                                 NASSP Masters 5003F - Computational Astronomy - 2009
Example: recently discovered exoplanets
            Planet 1
                                     Planet 2




   Credit: Gemini Observatory/AURA
                                          NASSP Masters 5003F - Computational Astronomy - 2009
        Example: group (cluster?) with Cen B



      Raw                                                                                  Smoothed




     Bright
  sources                                                                                  Dummy
subtracted




              Schroeder, Mamon and Stewart – in preparation.   NASSP Masters 5003F - Computational Astronomy - 2009
The other possible remedy:
             2. Try to reduce the
                instrumental
                broadening.




                      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.’


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




Credit STScI               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.
                           NASSP Masters 5003F - Computational Astronomy - 2009
                          Wavelet example
F Damiani et al (1997)




               Raw data                               Wavelet smoothed


   Multi-scale wavelets can be chosen to return best-fit ellipsoids.



                                                        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.
                                     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.
                                     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.
                                     NASSP Masters 5003F - Computational Astronomy - 2009
P Kroupa (1995)
                     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?
                  P Schechter (1976)
                                       • Distribution for most objects
                                         has a long faint-end ‘tail’.
                                         – Schechter functions.



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




                                  NASSP Masters 5003F - Computational Astronomy - 2009
             Relation to logN-logS
• Just as flux S is related to luminosity L and
  distance D by
                  S α L/D2

• 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.)
                                  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).
                                  NASSP Masters 5003F - Computational Astronomy - 2009

				
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posted:4/14/2013
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