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ASTR 551, Majewski [SPRING 2018]. Lecture Notes

ASTR 551 (Majewski) Lecture Notes




References: Read Binney & Merrifield Section 3.6.

Other useful references are: Sandage in The Deep Universe, and Trumpler & Weaver (1953) Statistical Astronomy.

Statistical studies depending on data that have inherent uncertainties are subject to a number of biases that affect the interpretation of results.

These bias effects (not surprisingly) are most pernicious where the observational uncertainties are "significant" and/or when some kind of limits are imposed on the survey sample. Three particular effects are commonly seen in Galactic/extragalactic studies:

Eddington Bias

Reference: Eddington (1913, MNRAS, 73, 359), Trumpler & Weaver, pp.123-126.

In astronomy it often happens that we make count distributions of objects having successive values of a certain measured property, for example

In general, the observations have observational uncertainties. Let's assume:

These uncertainties must have some effect on the data.

Following Eddington, we will explore the situation using the specific case of starcounts, but the same formalism holds for other measured quantities.

After working through the math, one finds:


(Top) Some observed distribution function Ao (in the starcount example we have been doing, this is Ao(m) ). (Middle) The first derivative. (Bottom) The second derivative, which is used as a multiplicative factor in the correction of the observed function in the top panel, to give the true distribution function, At .

Malmquist Bias

Reference: Malmquist (1922, 1936), Binney & Merrifield, Section 3.6.1. Sandage in The Deep Universe. One of the more troubling effects in surveys of stars and galaxies is the Malmquist Bias.

If we were ideally/properly determine density laws, D(r), or luminosity functions, Φ(M), we would use volume-limited surveys -- i.e. sample every star/galaxy in the survey volume.

As a simple example, imagine a sample of of objects with an even distribution of brightness, Φ(M), from Mbright to Mfaint, and ignore reddening:

Again, the net effect of the flux limit is that more luminous objects will be over-represented.

We want to calculate the correction one needs to apply to go from an observed sample mean absolute magnitude, <M>m, to a true population mean absolute magnitude, Mo if an mlim is imposed.

Case Study: Naked Eye Stars

A rather dramatic example of the Malmquist bias is one you should already be familiar with:

The absolute magnitude distribution of naked eye stars is dominated by very bright, evolved stars, even though these are much less represented in the true local luminosity function than faint stars.
Case Study: Gravitational Attractors

In our discussion of the local universe, we found that large mass concentrations can cause flows of galaxies that appear to have peculiar velocities deviating from the Hubble law.

Recall the Great Attractor, whose presence was inferred by non-Hubble flows, presumably due to infall of material to the large mass concentration:

Some controversy about the GA came about due to concern over the effects of Malmquist bias.

Lutz-Kelker Bias

Reference: Lutz & Kelker (1973, PASP, 85, 573)

Binney & Merrifield, Section 3.6.2.

The latter equation shows what the form of the correction function implies. Two factors:

This means that (see Lutz-Kelker Fig. 1 below):

  • Note that what matters is NOT the absolute size of the error chosen to limit a study, but rather σ / π .

    Case Study: HIPPARCOS Satellite

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    All material copyright © 2003,2006,2008,2018 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 at the University of Virginia.