NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 7 Confusion Dynamic range Resolved...

NASSP Masters 5003F - Computational Astronomy - 2009

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


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 powerto make the plots look better.)


A 1D simulation.• Add in instrumental

broadening.


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…?


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.


Near-confused fields:

A NICMOS exposure towards thegalactic centre.

Credit: Spitzer Science Centre/STScI

An all-instrument mosaic of XMMEPIC cameras. A rich stellar cluster.


Two possible remedies:1. Subtract sources,

starting with the brightest.– Eg the CLEAN

algorithm in radio interferometry.

– Eg 2: sExtractor.

Brightest subtracted


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 rangeonly 104 (=40 dB).


Example: recently discovered exoplanets

Planet 1Planet 2

Credit: Gemini Observatory/AURA


Example: group (cluster?) with Cen B

Smoothed

Dummy

Raw

Brightsources

subtracted

Schroeder, Mamon and Stewart – in preparation.


The other possible remedy:2. Try to reduce the

instrumental broadening.


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.’


Eg the Hubble Deep Field.

Credit STScI


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.


Wavelet example

Raw data Wavelet smoothed

F Damiani et al (1997)

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


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.


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.


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.


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)


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)


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/D2


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 Lecture 7 Confusion Dynamic range Resolved...

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