Malmquist bias

The Malmquist bias is an effect in observational astronomy which leads to the preferential detection of intrinsically bright objects.

So, the amount of light per unit of surface area of the sphere (called flux in astronomy) decreases with distance and therefore time.

However, magnitude limited surveys are the simplest to perform, and other methods are difficult to put together, with their own uncertainties involved, and may be impossible for first observations of objects.

As such, many different methods exist to attempt to correct the data, removing the bias and allowing the survey to be usable.

Unfortunately, this method would waste a great deal of good data, and would limit the analysis to nearby objects only, making it less than desirable.

Of course, this method assumes that distances are known with relatively good accuracy, which as mentioned before, is a difficult process in astronomy.

The first solution, proposed by Malmquist in his 1922 work, was to correct the calculated average absolute magnitude (

If the spatial distribution of stars can be assumed to be homogeneous, this relation is simplified even further, to the generally accepted form of The traditional method assumes that the measurements of apparent magnitude and the measurements from which distance is determined are from the same band, or predefined range, of wavelengths (e.g. the H band, a range of infrared wavelengths from roughly about 1300–2000 nanometers), and this leads to the correction form of cσ2, where c is some constant.

Unfortunately, this is rarely the case, as many samples of objects are selected from one wavelength band but the distance is calculated from another.

When this happens, the variance is replaced by the covariance between the scatter in the distance measurements and in the galaxy selection property (e.g.

[7] Another fairly straightforward correction method is to use a weighted mean to properly account for the relative contributions at each magnitude.

The second complication is cosmological concerns of redshift and the expanding universe, which must be considered when looking at distant objects.

If z is the object's redshift, relating to how far emitted light is shifted toward longer wavelengths as a result of the object moving away from us with the universal expansion, DA and VA are the actual distance and volume (or what would be measured today) and DC and VC are the comoving distance and volumes of interest, then A large downside of the volume weighting method is its sensitivity to large-scale structures, or parts of the universe with more or less objects than average, such as a star cluster or a void.

[10] Having very overdense or underdense regions of objects will cause an inferred change in our average absolute magnitude and luminosity function, according with the structure.

This is a particular issue with the faint objects in calculating a luminosity function, as their smaller maximum volume means that a large-scale structure therein will have a large impact.

Brighter objects with large maximum volumes will tend to average out and approach the correct value in spite of some large-scale structures.

Each of these values have their own distribution function which can be combined with a random number generator to create a theoretical sample of stars.

The reason that this estimator is useful is that the inverse regression line is actually unaffected by the Malmquist bias, so long as the selection effects are only based on magnitude.

[13] Advanced versions of the traditional correction mentioned above can be found in the literature, limiting or changing the initial assumptions to suit the appropriate author's needs.

This gave a much more exact and accurate result, but also required an assumption about the spatial distribution of stars in the desired galaxy.

[14] While useful individually, and there are many examples published, these have very limited scope and are not generally as broadly applicable as the other methods mentioned above.

Some alternatives do exist to attempt to avoid the Malmquist bias, or to approach it in a different way, with a few of the more common ones summarized below.

[5] Clearly, in this case, the Malmquist bias is not an issue as the volume will be fully populated and any distribution or luminosity function will be appropriately sampled.

Rather than trying to fix the absolute magnitudes, this method takes the distances to the objects as being the random variables and attempts to rescale those.

The homogeneous case is much simpler and rescales the raw distance estimates by a constant factor.

Unfortunately, this will be very insensitive to large scale structures such as clustering as well as observational selection effects, and will not give a very accurate result.

In both cases though, it is assumed that the probability density function is Gaussian with constant variance and a mean of the true average log distance, which is far from accurate.

As light rays leave the star homogeneously in every direction, they fill a sphere whose radius grows with time. As the sphere grows, the surface area increases; however, the number of light rays stays the same. Following a patch on the surface of the sphere with area A , fewer light rays pass through that patch as the light travels further from the star.
In a volume of space filled with stars, the stars have a range of luminosities and have some average luminosity shown by the dashed blue line. However, the more distant, less luminous stars will not be seen. This lowest luminosity that can be seen at a given distance is depicted by the red curve. Any star below that curve will not be seen. The average luminosity of only the stars that will be seen will be higher, as shown by the dashed red line.
The dashed red line is an example luminosity function when the Malmquist bias is not corrected for. The more numerous low luminosity objects are underrepresented because of the apparent magnitude limit of the survey. The solid blue line is the properly corrected luminosity function using the volume-weighted correction method.